Yeah, you can put horses on any pasture and they will happily graze on whatever weed grows there. Gryphons (just one example of a popular fantasy mount) should be carnivorous. So how many cows do you need to feed that thing before it will fly someone safely across the countryside, without deciding that this someone looks awfully tasty?

Big flying mounts should be horribly expensive to maintain. Only the most wealthy nobles and wizards who can literally conjure money should be able to pay for them.

Personally I always have to suspend a lot of my disbelief whenever dragon mounts are being mentioned in fantasy books and there is absolutely no explanations how these creatures are being fed. Usually the topic is quietly sidelined or simply dismissed with "something something magic".