I could perhaps live with having a character with a set race and class, though personally I would prefer to have a party that you created yourself.
Something that I would like to see, but would perhaps be really difficult to pull off well, is a system where you get to set the relationships between your characters, between friend, best friend, rival, sibling, mentor, parent, "business partner" and the like. This would have implications for how dialogues would work in the game. To go with this, I was thinking earlier that it would be cool if you could have parts of the game that split the party, having portions where you control fewer of those characters, or maybe a single one. And I would not rule out that decisions could be made that actually removed a character from the party, though I think I would make this sort of thing pretty transparent, like, " Which one of your will take my offer of power? All it requires is your companions life, " and " Neither of us, you bastard! " or " I will! " In the end game, you could have the party split up into groups or individuals based on their background elements and actions during the game, before coming back together for the final encounter or encounters. So, maybe you get teamed up with best friend, or parent, or rival. Or maybe the lonely rival gets teamed up with the parent, and they come to an understanding that helps the child and the rival get along together better, and so on. This would be difficult to pull off, but it is the sort of thing you could do to make a 1-10 game rich and really replayable. And multiple final endings? Totally. Wizardry managed to tie different endings into the same sequel. So could a D&D trilogy.