I think another problem is it would only make things worse with the exclusive mounts (ala, original 1.0 goobbue, Legacy chocobo, CE version coreul) would be either made pointlessly weak so as not to unbalance the 'normal' mounts that are freely available, or else be made 'special' and hence overpowered, and thus cause a ton of new bile aimed at players which have them. Not to mention the models would have to be redesigned to accommodate barding.
Instead of using the other mounts as companions, I'd more like to see SE add the option to actually use your mount in combat while mounted, FFT style. I mean, it wouldn't be too hard to program into the game - there is already mounted combat abilities in the game (during the Praetorium,
you have to use the magitek armour to destroy the robot arms that block your way, and hence extra commands to use the magitek armour's weapons are added to your hotbar temporarily.
Hence it wouldn't be too difficult to extend that further to all mounts. But I guess, balance probably has a lot to do with it.
Still, I can dream.