"They" cannot choose anything. Because there is no "they". There is one guy with a mount and four other players who don't have this restriction. Sure, the mounted guy would LOVE to avoid any situation that would prevent him from using the mount, but, since he's in the very minority position in the group, his voice doesn't carry all that much weight.
Which brings us back to the "don't play with spoiled 5 year olds" maxim. Particularly illogical 5 year olds who can't figure out that they, personally, will benefit from choosing options which benefit EVERYBODY at the table (because they will have greater resources and power to bring to bear on the problem).
If you do nothing but dungeon crawls, then, hey no problem. Mounted knights aren't a problem for you. Me, my worlds tend to be a bit broader than that. To each his own.
You can't claim that mounted knights don't work in your campaigns because the majority of your adventures take place in a dungeon AND simultaneously claim that the majority of your adventures don't take place in a dungeon.
You can continue to ignore the rest of the group all you like, but, unless you do nothing but lone wolfing, or the group creates an entire party of mounted character concepts, you'll run into the situation where the mounted guy is not really "mounted guy" but, "Vanilla Fighter who happens to get to ride a horse once in a blue moon".
Uh huh. Okay, I'll bite: Name me a single novel-length work of fantasy in which one of the characters is ALWAYS mounted. Never, ever do they leave their horse. Not once. Not ever. They never do anything unless their ass is firmly planted on an equine back.
At this point you've gone from rather silly to outrageous. You're clearly very wedded to the idea that "characters with mounts don't worK", and you're willing to say the most ludicrous things in an effort to make that true.
Your claim that the only way a mounted character works is if they get to shine MORE than their fair share of the time is simply insane. It's like claiming that fighters aren't playable because sometimes the thief will be picking a lock.