You didn't actually answer the question.
I did though, but I'll restate it again: The player describes what he or she wants to do. The player determines what his or her character does, thinks, and says. Those are the rules. What a character "should" do can therefore only be determined by the player. The player is not limited by what is on her or her sheet. That information is only there, in part, to resolve actions that have uncertain outcomes and a meaningful consequence of failure (as determined by the DM).
The game provides the player with specific options for action resolution, both physical and mental. How is going outside those options not a violation of those rules/guidelines?
There is no limit placed by the rules on your action declarations. The player describes what he or she wants to do. The DM narrates the outcome of the adventurer's actions, sometimes calling for a roll when there's an uncertain outcome and a meaningful consequence for failure.
That's not correct. 5e expressly charges the DM to call out and discourage metagame thinking by players. If an action declaration is metagaming, the DM can absolutely call it out and even stop it. DMG p. 235. A player consistently going outside the tools/abilities his character has qualifies.
Perhaps however you define "metagaming" or "metagame thinking" isn't how the game defines it. A lot of people drag in their ideas of what these terms mean from other games or how their older cousin taught them or whatever.
It's very clear what it means if taken on its face: Players are discouraged from making
bad assumptions based on out-of-game concerns that negatively impact the game experience. It gives specific examples such as as getting their characters killed because they don't think the DM would put a deadly monster in the adventure or wasting valuable time exploring a door simply because the DM was a little wordy when describing the environment.
That is the sort of "metagame thinking" they recommend discouraging by suggesting to the players they have their characters take action in-game to assess things. It says absolutely nothing about, for example, a player of the barbarian coming up with an action that someone else at the table thinks is not suitable for a low-Intelligence barbarian.
Playing through the character has nothing to do with player agency. It has to do with using the tools you chose/were given and the DM enforcing this. The DMG recognizes this by encouraging the DM to, for example, say "What do your characters think?" (DMG p. 235) when he sees the players acting outside of their characters.
"What does Plunk think?"
Whatever Gary says Plunk thinks. But Gary is well-advised by the DMG not to have Plunk act on bad assumptions based on out-of-game information that may hurt the party or the play experience. That's all the section you are quoting means.
The DM gets to decide the outcome of an adventurer's action. He or she doesn't get to decide what sorts of actions the player of that adventure may take, nor can other players. The smart play for Gary is to have Plunk do things Plunk is good at in the event the DM calls for a roll.
But the choice is up to Gary. If he wants to engage in the exploration or social interaction challenge the DM presents, he better hope the dice are on his side when the DM asks him to roll - and that perhaps he has Inspiration in his back pocket from all that excellent portrayal of the character he did earlier.