Turning radii would be neat, if technically detrimental... So that certain mounts don't look weird when turning around (I.e. Regalia. Also... How DOES the Regalia turn? Like, all the wheels are inside the bodywork, so unless it works like a Tank and drives only right/left side wheels it wouldn't actually be able to turn and if it DOES work that way, well, it'd have a trash turning circle and speed basically requiring it to stop and then rotate...)
Though, honestly, for "Feel" of mounts, I think BDO does it best. Not only does your horse (And carriage) feel like a physical being, with acceleration, turning radii, being weighed down by having a lot of heavy crap in the saddlebags, and different max speeds depending on the quality of the terrain... But they were also persistent entities in the game. You didn't poof them out of thin air. You picked it up from the stable and rode it to where you needed to be, then when you got off it... It'd be standing right where you left it until you returned (Unless you paid a nearby stable to come pick it up for you)
Obviously such a design would not work in XIV, especially not after having established the current system for so long... But it was really cool. (Even if I would be constantly asking myself "Dude, where's my horse?" while playing)
As far as improving mount abilities in FFXIV goes... I don't know... Since, it kind of limits the freedom of riding the cool mount you want, if you get a special bonus for having a certain ability in a zone (Especially if its a zone like Eureka where it's supposed to be butt to try and get around). It also opens up some "P2W" potential if MogStation mounts gained unique or better abilities. This is to say nothing about whether the engine can cope with adding such things in (We don't have any "Gliding" actions as is... We also need a loading screen to transition from above/below water)