Individual DMs will have to decide what their tolerance is for this, but having had long experience with this Variant rule in my games, players will absolutely try to add bonuses to the ability check the DM already called for if allowed to do so. It's effectively a retcon because you know you're now making a check and want to add to your description to mitigate the chance of failure.
This also creates something of an interruption of the flow of the conversation and should be the exception, not the rule, in my view. Compare Description-Call for Check-Check-Narrate to Description-Call for Check-Description-Q&A-Check-Narrate. It may seem like a minor thing, but this adds up quick to real time spent on back and forth hashing out whether something applies.