Strange thread.
To address the question in the title. How much control do DMs need? As much as it takes to run the game.
But the thing about power, hierarchy, control, etc that's often skipped over in regards to social situations is that the referee's power comes from the consent of the players. If the players refuse to cooperate, that's the end of the referee's power. At that point all they can do is exert social pressure or stop running the game. They have zero real power to force the players to do anything. Either the players cooperate, or there's no game. Likewise, the referee can refuse to cooperate...and that's the end of the game. Either the referee runs the game, or there's no game.
It literally doesn't matter what the books or rules say. Either the players go along, giving the referee the power to run the game or the players refuse, withholding that power from the referee, thereby making it impossible for them to run the game. At no point does, and it's not actually possible for, the rules in the books to have power over the players at the table. The referee and players at the table have to consent to use and abide by them. That's what makes the rules "binding", in as much as they actually are binding. We consent to follow these rules, therefore they have power...because we decide to give them that power. We agree to give the rules authority. (And we acknowledge that we'll inevitably argue endlessly about what the rules actually mean when they're poorly written.)
This is why it's possible to house rule things and cheat in RPGs. They're not computer programs executed by the meat-computer of the referee's brain. They're social constructs we consent to participate in and be bound by...only in so far as we consent to be bound by them. The referee decides to change the rules and inform the players, i.e. house rule the game. The players can object or dissent, but ultimately their only choice is to consent or leave. If the referee is adamant, that's that. The players might be able to talk to, argue with, beg, cajole, etc or they might not. Depends on the referee and players. The player(s) decide to "accidentally" roll too many dice, roll the wrong die, do the math wrong, etc but still report good results. The referee can only point to the social contract of the agreed upon rules, but the books have no actual power to prevent anything. If either side feels the other has violated the social contract, the social construct of the game breaks down. But at literally no point do the rules in the book (or those you've agreed to) have the power to stop, prevent, etc any of this. The rules have literally only the power the referee and players give them...and only as long as they consent to.
All RPGs are, by definition, high trust games. You trust the people around the table to follow the rules you've all agreed to, no matter how light, heavy, mechanistic, or principle based. Without that trust, it's impossible to play. So people who've never played FKR games fretting about how they can't trust the referee "with that kind of power" is more than a bit paradoxical. You are required to have an incredibly high level of trust in your referee to play every RPG already. You trust the referee with an endless string of things, but having slightly more control over the mechanics of the game is a line too far. It's honestly weird.
The more I read and play FKR games, the more they resemble PbtA games without specified moves or advancement. You generally have a simple resolution mechanic followed by clearly stated principles and the referee runs the game based on those principles. Advancement is almost always diegetic. It takes about as much trust to play in an FKR game as it does to trust a D&D 5E DM with deciding what gives dis/advantage.
The rules cannot protect you from a bad referee or bad players. Because, again, the participants can always ignore the rules. Your only real recourse as a participant is to withdraw your consent. The social contract is all about deciding these things up front. But even with the social contract, tables still run into problems all the time.