How does completely decoupling Ability Checks from Abilities affect the game?

The PHB mentions how it is possible to make ability checks based on different abilities, a variant "Skills with Different Abilities" rule.

In some situations, though, your proficiency might reasonably apply to a different kind of check. In such cases, the GM might ask for a check using an unusual combination of ability and skill, or you might ask your GM if you can apply a proficiency to a different check. For example, if you have to swim from an offshore island to the mainland, your GM might call for a Constitution check to see if you have the stamina to make it that far. In this case, your GM might allow you to apply your proficiency in Athletics and ask for a Constitution (Athletics) check.

Some answers here in RPG provide a similar solution, and even provide a custom sheet which decoupled those.

I wonder how, in long-running games, implementing this variant rule for all skills (by using the aforemented sheet, for example) affects the game and its pace.

  • Does it slow down the game considerably as all players try and suggest which ability they use for something?
  • Do you end up with players trying to cheese EVERY check to use a specific ability "I climb the wall by leveraging my weight and the above counter-weight and choosing the simplest path, so I can use Athletics (Intelligence)"?
  • Does it encourage players to RP more, which despite the slower game pace, makes the game enjoyable (for tables that like RP)?