What some are calling "goal and approach" is really the player being explicit about what they are doing and hoping to accomplish. This makes it easier for the DM to adjudicate the action. It stands somewhat in opposition to players just asking to make ability checks without much else in the way of description. There is also nothing in the rules that supports players asking to make ability checks. (Though previous editions of the game
do support this.)
It strikes me as odd that anyone would argue
against players being more explicit in their descriptions, provided the player isn't being overly verbose or ponderous. Which is why you'll see me suggest "goal and approach" be tied to being "reasonably specific" and "succinct." There is also rules support for players being reasonably specific as to what they are doing, particularly as it relates to hidden objects. The player's approach matters for the DM determining automatic success or failure or whether some kind of roll is appropriate.
This specific aspect of play does not require any particular "playstyle" as you point out in your post unless that "playstyle" is opposed for some reason to players saying what they want to do and hope to achieve. What happens in threads like this though is examples are provided that end up bringing other aspects of a DM/group's approach into the discussion which is then conflated with the above.
In this particular thread, the OP was examining the meaningful consequences for failure for certain actions. "Goal and approach" helps with this because the player being explicit about what the character is doing and hopes to achieve makes it easier to see if the proposed action has a meaningful consequence for failure. If it doesn't, then there is no ability check. That an ability check cannot be called for without a meaningful consequence for failure (among other things) is not a "playstyle" though. Those are the rules of the game, not something that
@Ovinomancer,
@Charlaquin or myself simply made up. What constitutes a "meaningful consequence for failure" is up to individual DMs. And, of course, nobody is required to follow the rules if they don't want to.
What is probably long overdue is for someone (not it!) who associates their approach to playing with both "goal and approach" and other specific aspects of play to lay this all out in a single thread for reference later. Principles, techniques, etc. plus any rules that support it. The thread can then be bookmarked for reference when the matter inevitably arises in subsequent threads as different DMs discuss how they'd handle this game situation or that.