The way you phrase your question suggests that you have a bit of a misunderstanding of two fundamental pieces of the 5e rules. They are very frequently misunderstood because they differ from other games and previous editions of D&D in subtle ways. Having them clarified should alleviate the worries you have concerning this variant.
First is the roles of the DM and the players. The introduction of the Player's Handbook (which contains a surprising number of essential rules) explains "How to Play" D&D 5e as three steps.
- The DM describes the environment. This step is purely narrative and doesn't usually involve any mechanics of the game such as checks, classes, DCs, or anything else from the rulebooks. The DM relays what the characters see, hear, smell, feel, and taste.
- The players describe what they want to do. For the players, this step begins as purely narrative, which is where I think your worries are addressed. By the rules, players do not ask for checks. Players describe what they want their characters to accomplish. The DM then decides if any dice rolls are needed, and if so, what kind. A player might, after the DM calls for an ability check, suggest a modification to the check, such as adding proficiency or applying a class feature, but it is clear in the rules that the DM is the one that starts that process, not the player.
- The DM narrates the results of the adventurers’ actions. After dice are rolled, if rolls were even asked for by the DM, the DM describes what happens as a result of the narrative course of action described by the players. This is also a narrative step for the most part, describing what, as a result of step 2, has changed in what the characters see, hear, smell, feel, and taste.
So how these rules differ from what you described in your question is that players never ask for an ability check. They can try to narrate what their character does in such a way as to suggest a certain ability check or request a change to what the DM has asked for, but it is always the DM that asks for checks.
Second is the nature of Ability Checks. D&D 5e is ability driven. Almost every single roll of a d20 is tied to an ability. When a DM decides that an action described by a player for their character has a meaningful chance for failure and calls for a die roll to decide the action's success, the first and most important thing they decide is which of the 6 abilities are involved. Only after that is determined do things like the character's proficiencies in skills, tools, or weapons come into play. Now, there are plenty of times that the choice of ability and what proficiencies apply is either so obvious or is explicitly defined by the rules that the DM often glosses over this step. Attack rolls and saving throws rarely require decisions in this matter and the DM can assume that the players are on exactly the same page as them and skip thinking or talking about it. The variant rule that is the subject of this question, however, brings this decision process to the forefront. Thinking about skills first then attaching an ability to it will lead to the delays and cheese that you're worried about, while focusing on the ability, with proficiency with a relevant skill being a secondary add-on, will speed up decisions and encourage good role playing over cheese, especially when using this variant rule. A review of chapter 7 of the Player's Handbook might help here.
Now, this is what the rules say, but the rules also encourage you to change them to suit your group. Maybe your players like asking for checks. Maybe what you consider "good" roleplaying is better facilitated by a change. Great, change that rule for your game, but do it with the understanding that the variant rule assumes that you're following the rules exactly and you'll need to adjust accordingly.
In my experience, using the variant while strictly keeping the rules talked about here in mind has led to only positive changes to my games, and I strongly encourage its use.