So, just for clarity, I'm assuming that by "The other" you mean the practice of having the player make a Stealth check as soon as they declare their intention to try to hide or to move stealthily. And you somehow think that that can lead to the player "retrying". Also, IIRC it has been asserted (either by you or by someone else arguing along the same lines) that if the player doesn't get to "retry", then the DM must be constraining the PC's actions.
That would be a correct assumption.
1) I'm not sure if it was you, but previously we have been treated to a high degree of scorn for bringing into the discussion the assumption that a player might do something that could be viewed as playing in bad faith. A player making declarations to try to game the DM to give them 'retries' seems like pretty bad faith.
Wasn't me, although I do think it's better to assume good faith from one's fellow gamers whenever possible.
2) But more importantly, this would seem to require that either a) the player is calling for checks, or b) somehow the DM has been relieved of the authority to determine how and when to invoke the game mechanics. At my table at least, the players do not call for checks, and, if I have asked for a Stealth check 'in advance', as you might see it, the player can have the PC do anything they want and I will decide when and if another Stealth check is to be made. I suppose that phrasing might sound a little authoritarian, but that process seems to me to be solidly within normal DM purview and not at all remarkable. Is there something about that that is unusual or that you consider undesirable or controversial? If not, then why bring up this nonproblem?
I see what you mean. So, my comment about a player asking to try again was more referring to games in which players can ask for checks whenever they like. In such a game, if the player got a low result on a stealth check and asked to make another and the DM said no... I would find that pretty dissatisfying as a player. But "goal and approach" sidesteps this particular problem, which is one of the reasons I like that style. It sounds like at your table, like at mine, the players are not asking for checks, but rather are describing actions, and you as DM are calling for checks when you deem them necessary to resolve the actions the players describe. So, yeah, this would prevent the player from seeing the low result of a stealth check and saying "I roll again."
All that said, BTW, even though I am in the habit of calling for Stealth checks 'prospectively', I actually am finding the at-the-moment-it-matters approach appealing. I kind of hope there is additional discussion of how to finesse the metagaming issue that doesn't bother you, because it does bother me, mainly because it would bother me as a player.
Assuming we are using goal and approach framework, and the question is just one of when to call for a Stealth check, there's the potential for the player to gain access to out of character information either way. If you call for the check "in advance," the player might see that they've rolled low, realize that they are unlikely to go unnoticed if there is something that might detect them, and decide to change tactics. If instead you call for the check only when it is relevant, this is no longer a problem. It does, theoretically, introduce a new problem where, if you later call for a stealth check, the player will know that there's something nearby that might spot them.
So, if you are concerned with "metagame thinking," either approach presents a problem, and it's a question of which you find more problematic. I would argue that the latter is less of a problem because the result of metagame thinking in that scenario is that the character either starts looking for creatures that might be looking for them, or shores up their efforts at hiding. Both things I don't think it is unreasonable for the character to do, whether they are aware of the presence of other creatures or not. The result of metagame thinking in the former scenario is that the character starts trying to hide, and then changes their mind for no clear in-fiction reason.