Haha. You stated all the things I did re the falling off the mount. :P On the forums on ENWorld, when I first saw that feature I laughed my head off picturing that. Knights in a tournament, fully armored, knocking each other off horses and always landing on their feet. Ridiculous :)
I like yours, but again, we have to be careful of double-ups. (Not so bad vs the cavalier one - however, we probably need a new name now :() I agree on not giving the feat, b/c then these guys have something to aim for if they want to be an even better rider (and will have advantages over other PCs that just take the feat).
For me:
Possible Names. Bonded Mount, Mounted Expertise, Natural Rider. (Name would depend upon what you go with below).
Adv vs Dismount. Keep
Adv on Skill Check
The skill part is hard, as you use Dexterity for staying on mount, including when it is frightened etc. But skill Animal Handling says all other things. There are also many parts to it.
For eg. Is this only when ON the mount? If so needs to be clear.
Or do we want them to choose 1 bonded mount (which they can change following a short rest with a new one - need to find an ability that works like this for wording.)
OR Expertise like a lot of class features: Proficiency in Animal Handling AND you always add double your prof modifier. (Again - I would find a feature that does this for wording). Again, need to decide if mounted only, bonded animal, or all animals.
Speed to Mount
Given the rules state it is a movement thing, I would definitely go with the cavalier approach and reduce the speed reduction to 5ft. of mvt. (Rather than half).