Having recently played with such a player I wholeheartedly agree with this. He was frequently so adamant that because he said it first and had rolled dice then he got to do it 1st, and the expression on his face when doing so was one of defiance and aggression - akin to 'This is what I am doing, I'm doing it NOW and don't any of you stop me.' He hated it when the DM stopped him and told him to let others have their say.
There is a very good reason the intention of 5E seems to be one of - describe an action, DM decides difficulty and suitable skill - and this is the 'help' rule. It gives other players a chance to say - 'I help you', and therefore grant advantage if applicable.
Another aspect to consider, if that if a group get into the habit of properly describing their actions it gives them *more* control over what their characters do. If they are vague then the DM might well make assumptions they do not like. *SPOILER* . Take as an example a famous trap in a famous dungeon that states that something nasty happens if an altar is touched by living matter.
Clearly - 'I gently check the altar with the tip of my dagger, looking for cracks, and maybe a secret compartment' is a perfectly safe thing to do. As is 'I poke the altar from a distance with my 10' Pole'.
But 'I run my fingers around the edge of the altar looking for secret compartments' is not, nor 'I try to push the altar', or 'I stand on the altar'.
As DM, I don't want to have to ask 'How do you examine the altar', as no matter how many times I say that, the players always get nervous. I want the players to be precise in their descriptions from the get-go.
Plus, if they are imprecise, I would assume 'I check the altar for traps', or 'I check the altar for secret compartments' means they are using their fingers to do the checking. Especially if they roll as they are saying it, thus giving me no chance to ask for more detail...
Asking for better descriptions *gives* more player control over their actions. More 'agency' if I'm forced to use that irritating buzzword.
RE the bold - i would agree completely for that for a game in which there is not a large body of experience bwteen the players and GMs for common actions established in their games. if the players has had the character search many many times there is, IMO, no really good reason to make them repeat the same phrasing every single time, any more than they should have to (as was fun in older edition) re-state from anew their "marching order" or "door protocols" etc over and over every single time.
As for whatever altar you are describing above, i would not likely run into the same problem you seem to have with the above because i would allow the results of a successful search check to reveal something was seriously amiss, in other words, a successful search gives them into that touching the altar or doing their standard operating procedure is BAD.
See, were i to put that altar in the scene, it would not be there to be used as a "phrase-trap" where its going to burn a player who says the wrong thing when it comes to description of action. Smells way too much like "you did not say you looked up." from many days of old.
Instead, on a successful search check their character would get more info and that info would inform that character that "touching that altar looks to be a bad idea."
For example, maybe they find lots of dead bugs right around the altar and tracks/bones that show that things which touched it died, other things tend to move around it at a close distance etc. maybe they also see physical but inanimate objects resting on or against with no problem, etc all of which can lead to "it looks like touching the altar is very bad, for living, not necessarily inanimate."
This kind of skews into my "the character is the expert, not the player" bent.
So, unlike how you describe the resolution, to me its not "you can get burned if you say it the wrong way" but its "you can get burned if your character fails to find sufficient info to tell them "dont do what i normally do."
makes "burned" more a case of "character failure at task" than "player failure at wordsmithery."
Whether or not those are issue for a given Gm or game or players is another thing altogether.
But, in a game like you describe, i could have, say, a dozen or so pre-printed index cards with very detailed "game-tested for this GM" wordsmithed responses for "how do i..." and pull one and read it each time to meet your established and consistent standards... but that does not seem as much an add to the game to me as assuming the character's skills, aptitude and quality of effort (check) is determinant of the consequence of the action.
To be very clear... in games i have run, played in and observed...
1 - i have seen among players more cases of "negative reaction" to consequences for PLAYERS failing "wordsmithery checks" ["you said it this way so... BAM"] than i have for characters failing skill checks ["Your characters attempt failed so...BAM"].
2 - I have also seen worse "outcomes" in general terms as far as of "lessons learned" from the "player wordsmithery" side those two approaches. The former tends to "teach" players to be wary of what they, the player, say. tends to lead to more "wish-proofing" sort of focus and efforts in play because they know their slip of the tongue can be a "gotcha". On the other hand, if they believe success failure lies as much or more with their character's aptitudes than their phrasing, they tend to focus on the character in play more than proofing their player to Gm phrasing.
But clearly, that is not necessarily going to be the same for everyone.