Not buying it for the vast majority. You have rules in place that govern the terrain impairment, opportunity attacks and climbing being slower etc etc. and it generally takes more than an ability check to change that. And the flyer avoids most all those things. without any rolls because of his plot coupon.
Climbing is a little bit difficult in that Thieves basically have a percentage chance to climb so if we give a Fighter a climbing skill then we would have to give the Thief something to make up for losing their thing. Maybe they just get a flat bonus on top of their stat check.
I dont know that we need to do anything about opportunity attacks. If you want to climb then you take opportunity attacks because you are not fighting the same as if you cast then you take opportunity attacks because you are not fighting. Fighters at least will have both the AC and Hp to deal with opportunity attacks
Well for one he might not be able to make ranged attacks while doing it.(and possibly the round after for that trance like focus )
He might be spending a hit die/healing surge to simulate extreme fatigue or pulling muscles on that super leap to get out of the way of the ranged attack spell. (this might be one expenditure for the stance duration - if he was actually attacked by a ranged opponent)
Another possibility by focusing on ranged defense - you might grant advantage to melee attackers. - might even only give disadvantage to ranged opponents but that is weaker than flight.
If one is comparing to a flier is giving up his ability to attack entirely then the fighter might be in so defense focused mode unable to attack in effect he is using his attack action to put up that wall of steel or to make a surging leap.
I have a character that can turn into a raven in raven form he cannot really attack but he can fly all day that way and it also acts as a superb disguise.
And the melee combatant might need to be able to confer many of these advantages on an adjacent ally as flight spells often allow it. (throwing them from an area of effect guiding them over difficult terrain with no loss of speed etc)
Additional mostly flavor things.
For the paladin/oath bound hero he would have an oath against making ranged attacks. And i suspect most melee oriented fighter might not have great ranged attacks anyway. So they are giving up a non-favored attack like a mage giving up melee attacks.
If we take all of these abilities which over lap a lot of the Thief and Monk abilities, I suggest decreasing HD to a d6 but keeping the same Fighter Atk, Saves, Weapons and Armour.
Of course since this is not talking about balancing against specific flight ability.
How can we balance the Fly spell when the Fighter gets a Griffin mount? All the benefits of Fly plus extra attacks and it only costs gold and can be used the whole day.