Here's a question:
It's understandable that some people wouldn't like the Battlemaster, same as any subclass. Not everyone would care for the expendable dice mechanic, or maybe they'd see it as "melee spells" similar to the 4e power list. Fair enough.
But, if that's the case, then why would you want a second/third subclass that functions the exact same way? Just instead of "Battlemaster Maneuvers", you have "Cavalier Maneuvers"...either pre-selected or possibly drawn from a second list?
In theory I like the idea of having subclasses for different specialties. But I wouldn't want to see a bunch of subclasses created just for the sake of creating them, with nothing really new or unique to offer. If you like the Battlemaster setup, then a riding maneuver or something similar would help you get the flavor you want. If you don't, then having two new "Battlemaster" subclasses likely wouldn't be much of an improvement. You're left with the choice of Champion, Eldritch Knight, or three different "Battlemasters"...two with preselected maneuvers and one where you can choose your own.
My thoughts are that if you like the BM maneuvers, mechanics, everything else, then it's fair to leave this as the "build-your-own-Fighter" subclass similar to 3rd ed. The same way Champion is supposed to emulate the older Fighters from earlier editions. It might not be for everyone, but it'd be there for it's intended audience....3rd ed players and advanced folk who appreciate that granularity. On the other hand, if you don't like the BM for whatever reason? That's fine too. But since you don't care for it, then use this as an opportunity to get something completely different instead.
There are concerns that lumping everything into BM would make it too powerful, but it's still limited by action economy, number of dice, and maneuvers known. Adding more options would be similar to adding new Sorcerer spells...characters would largely feel different by how they made these limited choices.
The other big one is that, given the choice from a list, no one would choose sub-optimal maneuvers and these wouldn't ever be used in actual play. But if they're put into a subclass instead, no one would choose a sub-optimal subclass the same way you don't see many Land Druids or Berzerkers. Even Beastmasters are a little better off, as they have a stronger theme differentiating themselves from other Rangers. Similarly, without really strong fluff, people may choose to be a Battlemaster anyway and simply call themselves a mounted knight (possibly with the Mounted Combat feat and Knight background as substitute).
If it were me, I'd probably create Cavalier as a fighting style. That would be powerful and always available, given early in a player's career, and open to Paladins and Rangers who might like that theme as well. Since it's limited to only when mounted, it would have to be a little stronger than abilities that are "always on" like bonuses to attacks or AC. For Scout, there's probably room for this as a subclass...DEX-based, abilities to use instead of heavy armor, more direct and more martial than a Rogue. Less magic and less woodsy than a Ranger. Mobile, but not super-fast and unarmed and borderline mystic like Monk. Possibly some more non-combat utility than the base Fighter. Fits a nice niche.
If I were to use the expendable dice mechanic they're proposing, that's okay but I would change it to make it feel less like "Battlemaster without the ability to choose". I'd look to Bardic Inspiration for my example...the extra dice works similarly, but it's used differently enough that it feels like something entirely new. I do love the idea others have proposed for making them refresh on moving and shooting from cover...it's great when mechanics can align both what you can do and how you go about doing it.