PHB p 185: Roleplaying is, literally, the act of playing out a role. In this case, it's you as a player determining how your character thinks, acts, and talks.
So, yes, by the rules of the game the player is given autonomy - or full rein - over how their character thinks, acts, and talks. Of course, the results of those actions are left up to the DM to adjudicate.
Maybe that is what roleplaying is, but is any role allowed at a table ? Certainly not, again, at the very least I refer you to
don't be a wangrod. Being able to roleplay does not allow you to do whatever you want with that role. And there have been multiple examples of at least influencing how a character even thinks, for example through the use of charms. Not only is it expressly stipulated that, whatever you think your character might be thinking, he now considers a previous enemy a friend, and thinks of him in those terms, but it would be very bad roleplay to play the character otherwise. So obviously the citation that you gave us does not grant unsupervised authority to the player, even unto the thoughts of his character...
Rather than "your character cannot do that" - which, to me, is far too close to "your character wouldn't do that" - I'd say "that action is not possible because... reasons" and probably see if they wanted to take a new approach since it is very likely that their capable adventurer would know if something was truly impossible.
I'm sorry, but (and it happened to me 2 weeks ago), there were multiple holes through a wall through which a hydra's heads were attacking, I killed the hydra, but she was regenerating, so I said "I crawl through the hole" and the DM just told me "you cannot do that, the hole is too small, you cannot fit." You can say what you want, but I did not have full reins on my actions, and it was normal, and I certainly did not complain.
Also, they shouldn't have 'more power'. They just have a more expansive character in the entire rest of the world.
At the very least, they have a power that the players do not: "A Dungeon Master adjudicates the game and determines whether to use an official
ruling in play. The DM always has the final say on rules questions." So, obviously, officially and technically, they have more power. And after that, having in control an entire world is obviously more powerful than controlling a single individual.
After that, it's OK, because with more power comes more responsibility, namely the fact that they are running the game for the players' entertainment. But, absolutely factually, they have more power.