Would people stay on ground mounts if flying was allowed in battlegrounds?
No, they wouldn't. Even if it was their stern opinion that BGs are designed to be played on the ground.
Having to willingly gimp oneself in order to experience content as intended is terrible design.
It's one way or the other. Either you "force" people to experience content as designed, or you let everyone skip it because "choice". No one would fight through a castle when they can safely land on the tower where the boss is. Even if they would enjoy everything from voice acting, scripting and so on between the entrance and final staircase. Players are like electricity, they choose the path of least resistance.
If Blizzard aren't allowed to restrict players, then truly they might as well hand out gear and everything else from a vendor. Why not, people could choose to get them the real way right? Yeah, very likely indeed. And wonder how long they'd stay interested then.