Leadership advantage because the unit's leader is on a horse, and therefore highly visible to the men he commands. This helps with all battlefield command, control and communications really.
Horses could be just like captains: basically a name with a few game mechanics changes. Being on a horse presumably helps with leading cavalry. But it's plenty plausible to have one's horse killed from underneath you, with archer fire and loads of infantry. Every time your horse survives, you keep it's bonuses, maybe you increase your jousting skill. (I know you can increase jousting skill currently when leading cavalry. This would simply apply to whenever the character has a horse.)
You might find yourself valuing your horse more than your captain. And that's how things ought to be in Battlemaster-verse! Commoners are common; horses are pretty beasts.
And, being on horseback increases your characters damage output whenever that's calculated. For Heroes, I think? For Adventurers too but... horses ought to cost a bit of gold, and be purchasable as Paraphernelia. Adventurers normally ought not to get them (but maybe can steal horses...?)
I am really just tossing the idea of horses as relates to the medieval noble/knight person out there. Horses were pretty important, much like cars but also like battlefield weapons. There's a scene in Henry V where the French knights are all sitting around, bragging about their horses. We could RP that sort of thing here, but it depends on players agreeing what the horse(s) was(were) like when if there was a simple number in-game the only disagreement can be between the characters.
