5e already has rules for limiting what a character can do depending on strength score - Carrying Capacity and Push/Drag/Lift. It's reasonable that these rules are extended to certain strength checks. Otherwise, at its most extreme, a Cat (Str mod of -4) can succeed on a DC 15 Strength Check.
Also, Larian already implemented strength restrictions for throw - you need X strength to throw a character of Y weight, so this could simply be extended to some checks.
I would also argue that strength checks should be repeatable by the same character, assuming time allows. E.g., you can keep trying to knock down the Inn doors, but each failure progresses the fire more. And perhaps the strength check DC is reduced by 2 each time to represent progress made in weakening the door, and the soldiers automatically break it down if you fail 3 times. But at that point you can only save one of the two people in the inn; the other one burns.
