How often do players come to the table with a preconceived character and does that usually work out?
I've had the issue where it was expected for the game to bend around what they wanted to play and not for them to develop their character through play.
Is this common?
It's common for players to want to do it that way. If you can trust the internet, it's almost as common for some GM's to try to give it to them.
I've been more of a purist the other way--no player-written background is going to change my setting by one iota by virtue of being in the character's background or because the player wants it. I don't have any trouble getting players. They all seem to enjoy playing the way I run it. Of course, I have no way of knowing for sure about the ones who didn't like my games and how much that figured in it.
Plus, it is not as if I don't get player input. I love player ideas for generating a campaign or setting. I'd rather have them direct from the player when we are talking about campaign and setting. Character background strikes me as a singularly inefficient and ineffective means of making the entire group happy, when we can chat regularly during breaks about what we like in the campaign and what we'd like to do, then later I make a few notes. Not to mention that sometimes when you do it that way, half the players have no idea what they want until a player makes an off-hand suggestion and they all say, "Yeah, we want to do that!"
I have done it the other way: We've worked out what kind of setting we are playing. I'm in the process of developing it. The players are making characters. I might ask a particular player or the group in general if anyone wants a particular thing in their background, with the understanding it is relevant to the campaign. Sometimes they like the idea; sometimes they don't. Sometimes the player loves having a thing in their background that even that player doesn't fully understand, because ... they get to discover their background during play.
As for preconceived characters, I'm fine with a player having a preconceived character that fits the campaign we all agreed to play. However, I want the players to be
able to take any character handed to them and play it. It's a great skill to have and should be practiced occasionally, as ultimately I think it makes for better games. When we do one-shots to try something new, we'll sometimes use pre-gens. But other times we've had players roll up characters randomly, make the choices that weren't random, then throw the characters in a pile on the table and have everyone draw. It gets players to try something new. Most players (even mine) would chafe at this if you drag it out too long. My wife really doesn't want to play a big bruiser knight long-term, for example, if that happens to be the draw. She is game for a one-shot.