Sometimes gameplay has to take priority over lore and immersion and to me this is one of the cases where thats actually okay. There are mounts that people worked really hard for, but that would be "ground-only" using logic - all of the tank-mounts, Fenrir (the wolf, not the bike - MGP used to be harder to get...) and considering the T-Rex is locked behind Eureka, yep, that one too...
There is also quite a bit of development time that went into creating those mounts in the first place - mounts that would basically end up never being used because with the next expansion 3 out of 4 "gama-areas" will suport flying. Considering that ARR has much smaller zones than HW, SB and (predicted) ShB, I'd take a guess and say that you can probably fly in 80-85% of the game, so every mount that cant fly is basically useless to you most of the time.
On top of that you have mount-specific-music - and yes, yes, requests for a protable orchestrion have been made time and time again, but the truth is right now that we cant have due to how the code works so the best we've got are using mounts with the song we want to listen to. I like Ramuhs one. But I cant reasonbly use the Ramuh-pony right now, because well, it cant fly...

If immersion bothers you - and I can understand that it does or would - there are two things you can do:
1) Only use mounts that you think can/should fly and ignore the rest or use them on the ground only. Sure, you'll see other people use "wrong" mounts, but that doesnt happen very often, at least not to me... (maybe I'm to much at my house and to little in the open world...)
2) This world is pretty magical, so until they provide some actual lore why this and that mount can fly, you can always come up with your own reason. For example: I recall that the reason given for Fenrir being able to fly is that hes not actually flying but using aether/magic to freeze the air right below his feet and is "walking" on that thin layer of ice hes creating in the air. The Shiva pony could be doing something similar.
We have those floating jellyfish everywhere and if memory serves right they can fly because they store gas and some wind(?) aether in special bladders - whos to say that a crazy scientist in the alchemist-guild didnt come up with some sort of mixture that you can inject into the paws of a lion to achieve a similar effect?
Or you use the same explantion we use for the glamour-system and treat mounts that fly but shouldnt as an illusion someone has cast on, lets say their chocobo!

I'm all for lore-friendliness, but we all now that sometimes gameplay and personal preferences have to take priority and I very much feel like this is one of them.