Moreover, it's the GM's job to create things while it's the players job to interact with and change. If players wanted to create they'd be GMing - not playing.
Going by your analogy, it would certainly be the job of the players to carve and shape the material the GM sets for them as they see fit within boundaries, but it is not their job to add in new material just in an attempt to match what the GM has given.
To match it? Of course not, but to integrate with it, to expand it and perhaps even refine it? Absolutely.
I have to say, I've already become a fan of not having bios in character sheets for my games, or at least very short ones if I do. A bio is essentially a very unsubtle prompt for the player to explain who their character is, and who it most appeals to is people who like to create elaborate stories all on their own, even if they're mostly completely irrelevant to gameplay and even the plot in general. I like filling out character sheets with bios specifically because I get to write a story in them, and further gameplay doesn't quite factor into any part of it. It's why the 'deeper and more thought-out' games appeal to me despite the fact that the vastly overwhelming majority of them never go anywhere before dying off quietly. Writing the character sheet captures the exciting feeling of starting to write a cool story you had in mind (or, rather, what could be the beginning of a cool story you have in mind), and the guilt of inevitably failing to finish it when your enthusiasm peters out falls on the GM.
I am very guilty of this. Particularly when something else I'm building is in its death throes need a cookie. Which is the storytelling equivalent of binge eating chocolate cake because you wrecked your car, not healthy or particularly helpful. I always love bios, of any kind, but I'm becoming more and more enamored of the short and sweet ambiguous bios that I practically proposed marriage to a couple posts ago. I understand their dangers, but I think they are invaluable as an anchor for a character and as a guide for the DM.
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This. Honestly, I'd much rather find out about the character as the game progresses. Otherwise it's often just a wall of text contest, with the people putting the most syllables on paper getting in.
I believe that what we have here is a failure to communicate... Wall of text character entrees are, generally, bad. Elysium is the only game I know of that, for as long as it ran, that ran well with characters that had long bios. The thing in that game was that everyone had a decent amount of freedom to create their own little niche, and then everyone played their characters to the hilt.
In general, bios longer than a paragraph, maybe two, bad news. For an example of the worst bio of my career in writing bios, which I still grit my teeth over a little when I think of, look up the old Thawed game that GWG ran. The character I put in is a godawful piece of purple prose that has a tragic backstory that feels like it's trying to enter a competition for saddest life against a dozen Darfur war orphans.
This is a recent bio of the kind I've taken to liking,
A smiling young man with fair skin, dark hair that's clearly been cut with a dagger for the past several months, and grey eyes that seem to be perpetually laughing at something, 'Shaw is a self professed cheat, liar, and great admirer of women. Particularly friendly women in limited clothing and good mood lighting. Shaw would describe himself as an open book, but, whenever questioned about his family name, where he came from, and why someone with so few scars and such soft palms is running with a mercenary company, he'll almost always reply with something along the lines of "I am an open book, but I'm an open book with water damage, a bunch of missing pages, and good smear of jam in the binding that makes the chapters all clump together".
3-4 sentences, depending on how you count quotes. In combination with the names for his skills, the general picture of the character is pretty obvious. It's short, has a few decent hooks that could be used for a lot of different situations, and it demands very little of the world. Heck, the bio of one of the best characters I've had in my games, bar none, was two short sentences of bio, and six Hemingway sentences of physical description. It might have taken up three lines when put together.