I think you're right. But I don't think that it is a bad thing like you do. Being able to make a choice that is different than the one that you would otherwise be presented with can be a hard thing to do. Sometimes you have to be willing to risk what you thought you cared about to do the right thing.
And I think that your idea of the three (Harry, Ron, and Hermione) embodying the houses doesn't go far enough. Their is a character in the books that embodies Gryffindors and only Gryffindors: Neville Longbottom.
Harry is a natural leader. People just follow him. I mean, he's seen as a loser because we know where he is coming from but he's easily one of the most popular kids in his school and a star athlete. He is the embodiment of everything that Slytherin is for. Except he chooses. He chooses to stand up and do what he needs to do. He chooses to go against the grain.
Hermione (especially book Hermione) is an embodiment of Ravenclaw. She's smart to the exclusion of everything else. Almost to the point of being reckless. And that would make her a Ravenclaw... but she chooses. She chooses to use that knowledge in a way that helps people. Even when it risks her.
And Ron (especially book Ron) is a big time Hufflepuff. Everything about him is him trying to be a good friend. When someone gets bullied, he's usually the first one to do it (he stands up to Snape in the first book). When Harry needed a friend to help him out, even though it meant Ron would have to face the thing he feared the most (big ass spiders). He chose to do all those things.
And you have Neville. He's not smart. He's not popular. He's not a leader... except when he chooses. He was probably the bravest kid in the series, and not just because he stood at Hogwarts when the trio left...
Neville chose to stand up to the trio when they were being stupid in the first book.
Neville asked out Ginny before Harry could.
Neville was okay with fighting the boggart. And he beat it. He literally faced his fear with that one.
Neville faced Mad-eye's class without running even though he knew what the curses could do.
TL;DR: Yes, Gryffindors choose. They choose because choosing is hard and sometimes no one else wants to.