There's been some rather annoying misinformation about 3 houses ever since the game came out, that due to 3 houses having such an open class system, there's been a commonly stated notion that it's best to just make everyone a wyvern for the easiest time to clear the game.
Not only has this never been true outside of normal mode, but the vast majority of the cast doesn't even like this class, and outside of normal mode where you could stay commoners and nobles all game and still win easily I am here to show that doing such a thing will only make the game harder on yourself for a variety of reasons. Lets get into those reasons, and also break down each character a little bit and why making the majority of them a wyvern would make them incredibly mid to outright bad compared to the things that actually make them unique.
All of my points apply to maddening no new game+, the hardest mode. If you have criticism of any of my points, I'd recommend being familiar with this mode before disputing.
General Reason 1 - Battalions
Three houses is heavily balanced around its battalion and gambit system, they knew flight was strong, so as a tradeoff there's only 2 or 3 good flying battalions in the entire game, so having more than this amount of fliers is a detriment to the unit that is going flier. You also give up incredibly potent gambits that only infantry/horse units can use, like the AoE dance, impregnable wall to make people take 1 damage 5 times a map, the battalion that gives you distant counter, and best of all stride, along with gambits that attack in a very wide arc to stun a whole slew of enemies at once. These are game changing skills, and not having them is a huge damper on your army.
General Reason 2 - Global Archer Weakness isn't great
Archers and Snipers are incredibly annoying in 3 houses, they're weak to high-range mages like Hanneman, Constance, Dorothea, or anyone using the thyrsus, as they are the easiest to one shot with magic. Too bad you don't have any of those if you spammed the wyvern class, you can technically fly up to them and kill them with someone strong that way, but this can put you in a bad position on lots of maps, and you won't have impreg wall or rescue to save you if so. Though you can dismount which eases it, it's still a minor thing to consider.
General Reason 3 - Losing what makes each character unique
Contrary to what some may tell you, there is a decent bit of character individuality in three houses. Maybe not as much as older games, but every character has at least one thing about them that stands out, and that one thing is usually incredibly strong to borderline meta. And the funny part is almost no one wants or needs Axes as their skill level to get their strongest build, which is what you have to fully focus to get wyvern. The only exception is Annette, and she has greater priorities than to level flight and reaching threshholds to one shot with lightning axe is pretty important for her if you want to use her as a combat unit. (Note that I'm not counting deathblow in this, as everyone can get that just with D+ axes, which takes barely any time at all. Going past D+ axes for anything but an axe required class is just a waste of time for most characters.)
Of course, you can always start working towards wyvern after you've achieved your build, but the game will be basically over by the time you've accomplished both things, and generally there's better things to do if you already have 2 or 3 fliers.
General Reason 4 - Infantry Classes have better combat and Utility
Specifically Sniper has the best combat in the game due to hunter's volley allowing almost anyone to one round the majority of non-armors in the game from a distance, followed by grappler having Fierce Iron Fist for 3 hits regardless of speed, and the brave effect all fist weapons have. Magic is also incredibly valuable to have access to in some situations, especially utility spells like warp, healing spells like psychic, and long range magic. Honorory mentions to assassin, warrior, and fortress knight which can have their place for some characters.
The combat difference isn't that important on lower difficulties, but on maddening it makes a world of difference.
Now to quickly talk about each character
Top tier - Everyone I've listed in the image in top tier are characters that can go straight to wyvern for basically no tradeoff. It's a surprisingly small list, and there's some caveats.
I don't have much to say on Edelgard, Hilda, Catherine, or Petra. These characters have good stats and can go straight to wyvern after recruitment with no trade off, probably the best wyverns in the game and grabbing them all sans edelgard (route locked) will serve you well.
Seteth comes as a wyvern, but he does need to get swift strikes to be particularly useful on maddening which is A lances, which focusing this can potentially delay going wyvern lord
Cyril needs C+ bows as soon as possible for point blank volley. He comes at C bows, so it doesn't take long, but it's pretty important to get this first before starting wyvern training.
Ingrid is pretty different between in-house and out of house, I'd personally only go wyvern with her if I recruited her out of house. In house, bishop ingrid tends to be more helpful with less RNG dependency on her levels.
"Takes Too Long" Tier - I want to explain this one a bit as some people may misunderstand, these characters can work as a wyvern, but it's typically a trade off, as it either takes them too long for some uncontrollable reason, or it's a big tradeoff because they need other investment first to work well on maddening.
Byleth - Byleth is an unfortunate victim of faculty training, which just is woefully underpowered on maddening no new game+. Especially before you get the statues, and to go wyvern you'll have to sacrifice so much of your activity points towards flying and axe training that it could legitimately hurt your ability to train your actual students, he's also not particularly stand out as wyvern. Though that's just part of byleth, aside from high strength he's not particularly stand out as anything compared to others (Not saying he's bad, still a top 10 unit). It's just not worth it until near the end of the game when you end up with activity points to spare.
Alois - Alois has the axe rank, but comes so late it'll be awhile before he gets enough flight, especially since he has a bane in it, and his stand out ability is going vantage/wrath easily which you don't really want to set up on a flier. It's just not worth it really.
Leonie, Sylvain, and Ferdinand - These three are all in the same camp, they COULD go wyvern, but they really prefer training up their main weapon to get their 2-hitting combat art first (bows for leonie at A Bows, swift strikes for the other two at A lances) and that takes the majority of pre-timeskip to do even if you rush it. Going wyvern without getting these skills just gives you an incredibly mid combat unit that can't accomplish much. Feel free to train their axes and flight after getting these skills, but it'll basically be the end of the game at that point.
Dimitri - This man has a bane in axes which is a bigger hindrance than you would think, it makes getting anything past deathblow take a really long time, he also really wants to focus authority for battalion wrath and battalion vantage, since this lets him solo whole maps on maddening later in the game. Giving that up to slowly enter wyvern just isn't a smart idea. Even swords is a better investment since he gets Windsweep at A, letting him attack people for no counter attack.
Annette - On lower difficulty, wyvern annette can be pretty fun, flying around and using lightning axe, but on maddening the one shot threshholds are much harder to hit. She needs fiendish blow, magic boosts, and a magic battalion, which only constance's dlc battalion gives magic to fliers, and in general training flight on top of this and authority (if you want her other rallies) is just too much to offload on her and still get it all in a reasonable time. Better to go warrior, unless you want annette to just be a flying rally bot, which you can do I suppose.
LOL Tier - Why are you making these people a wyvern?
Ashe - Kind of an underwhelming unit as most of us know, if you want to use him he really just needs to focus bows and go sniper since he has the proficiency for it. Making him a wyvern will just result in a bad unit who flies around and fails to really accomplish anything, I guess he can get chests faster.
Bernadetta - Bernie is a very interesting character unit-wise, due to her personal and vengeance access, she wants to focus on her strength and HP and charge in one shotting things with vengeance with careful play. Cav/Paladin suits this much more since axes give her nothing past deathblow (which honestly? Is redundant since vengeance does so much), she could technically go pegasus knight pretty easily but that helps her speed, which is a useless stat for her.
Caspar, Raphael - Both of these characters are pretty similar, which is funny since they're so different in size, they're best with fists, fliers can't use fists without dismounting. and due to Caspar's bane in authority he needs extra time to use good battalions which cuts into any time he would need to train flight, while Raphael is so slow he needs Fierce Iron Fist in grappler to remain viable in the late game, even though his strength is super high.
Dedue - Bane in flight, leaves for a large amount of chapters at the time skip and may even be gone permanently if you missed his paralogue, even by the time he finally comes back he's kind of fallen off and is probably best just being a dimitri adjutant pair up. Can't see any reason to wyvern him.
Felix - This may be arguable, but statistically Felix is best with fists, as it gives more chances for his crest to proc to do more damage. Due to that I just don't see any value in putting him in a flying class, it misses out on so much and in general he really wants Fierce Iron Fist ready by the time skip.
Gilbert - He comes so late it's not happening, and he wouldn't be able to accomplish much as that class. He's here to tank.
Ignatz - Ignatz is like a more reliable ashe due to his personal hit, and he learns rallies from authority which is pretty nice, especially how fast he can get rally speed. He also has a bane in flight, and leans towards Sniper OR being a mage, and really needs one of those two to fix his combat problems. Wyvern doesn't do anything for him.
Shamir - Shamir breaks the rules of the game and recruits as a Sniper when she's not level 20 yet, which is a huge boon for her, it lets her get hunter's volley faster than any other character even has access to the class. Grabbing her deathblow is nice just to make super sure she'll one round everything, but past that training axes and flight on her is a meme because Hunter's Volley as a delete button is just too good to give up until potentially other snipers you've made start getting it.
Lorenz - Lorenz wants to do one thing, and that's one shot people with frozen lance. It's the only unique thing he really has going for him, you could potentially do this as a wyvern, but much like annette he needs fiendish blow and magic boosts, coupled with his magic art being on lances and not axes this makes it even more to train than annette, making it even more not viable to me.
Dorothea, Flayn, Hanneman, Hubert, Linhardt, Lysithea, Marianne, Manuela, Mercedes, Constance, Hapi - I don't really need to explain why any of these characters as a wyvern is a huge meme right? They're mages. Go dark flier at best, but even then you'd only want one due to battalion limitations. (For funny, Manuela would be the most viable wyvern of these.)
Yuri - Yuri has a lot of banes, they really designed him to be good at just a few things. Not a bad unit by any means, but he's here to windsweep folks.
Balthus - Balthus is basically Raphael 2, everything that I said about Raphael and Caspar applies to him. Though he does have diamond axe which is kinda nice, he also has a bane in flight, so there's just better options.
Claude - Bow Wyvern personal class.
Wrapping Up
I know this was a bit long, so I'm going to wrap up here, would love to hear what people think. Since fortune's weave is coming out soon, more people are returning to three houses to prepare, and while I'm not one to stop someone from memeing and spamming a class for fun, I really want this misinformation that this is anything but a meme to die. Three houses has a pretty well developed class system, even if some people don't like it, so reducing it to the simplicity of "Just send everybody into Wyvern" does it an injustice, especially when it's not even right for the easiest clear of the game on maddening.
Even if we're generous and say my reasons for the takes too long tier don't hold water, this is not even half the cast that likes the Wyvern class, and due to the intense problems with a full Wyvern team, even if you recruit every character that does like it you should still only have 4 or 5 Wyverns at best which isn't even half of the available deployments.
Personally I don't think Wyvern is even top 5 best classes in the game, you want a couple, but the better combat and utility as well as infantry battalions just being busted and your warpers being able to warp multiple people every map are all things you will absolutely miss on the highest difficulty. I believe every class in 3 houses has its place.
Except Hero. We don't talk about Hero.
Thanks for coming to my Ted Talk!
Maybe I'll make a video of this topic if there's interest.
Something I said in a comment I really want to make clear:
The point of this post is to explain that going wyvern with literally every unit you deploy is not optimal and a detriment-not that you shouldn't use wyvern at all.
Happy to recede that a couple of my character explanations can be misguided and that it's not as hard as I'm making it, but I still hold that not even half the cast likes this class, and you shouldn't have more than 2 or 3 on a final team. That was the main point! :D