What if one or more of the players came up with the idea or basis of the fictitious game, but another person offered or was chosen to GM it? You see the GM as more than just "a referee" then I take it?
The DM is effectively the game. Chaneg the DM, you got a different game. Don't like the game, find a different one. Part of this is supply and demand, part of this is the cold hard reality that every DM is different and they are not hot-swappable.
This is kind of like a director of a movie. It's a vision thing. Change directors, get a different movie, even with the same original script.
Another variable is "persistent pesty child syndrome". If a kid is going to keep asking for Mount Splashmore, I'm inclined to be more resistant to take them there the more they ask. Players get to say what cardinal direction they want to go, or who they want to parley or fight with. They don't get to tell me that the next villain has to be a vampire, or that they're tired of vampires despite the fact that they let the master loose and he's been making more ever since. The world moves where it moves (partly because the DM says so) and the PCs live there. The players don't get to demand what they want to happen next, their PCs will have to do some work.
To cover some of what I consider obvious, yes a DM will listen to what the players are interested in. If they want to try a murder mystery, he'll probably make one happen. A players are free to stop hunting down all those vampires, but there are obvious consequences that tend to keep the PCs stuck doing vampire patrol. Just like real life, some situations keep people stuck working, instead of pursuing their own goals. What I'm talking about in the previous paragraph is that players do not get to create game content or change. They can suggest or influence the DM. Their PC can try to enact the player's desire on the player's behalf. But the PLAYER does not have any right to direct control.
Ahhh. Now this is an important factor and certainly shows our angles of approach to this question. As for myself, I hate games with levels (and pre-defined class packages).
classes and levels are not tied together. Yes, D&D is a class and level based system. Shadowrun is a skills based and non-level based. It could have been skills based and level based (gaining a level lets you increase skills). The level acts as a metric, for the DM.
This is like folks who say they don't give out XP, they just say the party leveled up. In which case,they effectively gave the 3e PC 1000 times the level in XP. Same diff.
However, the foundation is the same, a new unskilled PC is likely young...
Then you would not accept a young character claiming in their backstory that they are a child and heir to the current ruling family (or son/daughter to the current Senator or whatever government the game uses), whose skills taken at character generation reflect?
I'm not seeing the contradiction in what you said, as it relates to what you quoted when you said that. Being the son of somebody important may justify having ranks in certain skills, but the PC can't break the rules on allocation...
However, I think you may be changing the subject to "what if the PC says they are an heir"? That's OK, let me consider that.
As a DM, when I see that, it's a warning flag. Is the player trying to weasel in some money/power/position? However, as a player, one of my longest running PCs used that tactic. So I'm guilty of doing it. In my case, I worded it as "son of councilman" and "venturing into the world to gain experience to be worthy of joining the council".
This was under my more permissive GM friend (not the bad GM). I was running the first PC in his new campaign world. What I had done in my backstory was lock in the elven nation, we were now a council of lords, fairly big (300+ members) and member ship was granted by reaching 10th level. I'd invented the seeds of an idea that the GM liked, so he filled in the rest. What I'd also done was set myself up as "sure my dad is somebody important, but I'm not getting any extra favors from it right now" as well as set up a goal (reach 10th level, become a council member).
There's a couple of points there. With that PC's 1 paragraph backstory, I just violated the strictest sense of what I've been talking about here. I made a goal for a 1st level PC and I invented a bunch of game world details.
I think the limits I see a DM placing on backstories are more of guidelines. There's a lot of grey area. Some players can make a backstory idea that the DM just loves because it fits. Other players come across as trying to pull a fast one. As a DM, limits and strictness are in place because of the latter type of player.
I think also, for a player, there's a desire to write a backstory that doesn't define your PC as shmuck. Some players go to far, trying to be "son of the king", and then when the GM twists it down to fit the PC's actual starting circumstance, the PC becomes a shmuck, because it's the only way to justify the son of the king NOT having all sorts of cool advantages.
So in some ways, the limits I'm talking about are also guidelines to players for how to avoid the DM from twisting your backstory so you're a loser. The first step is to not overreach.