That's rather a separate issue with its own solutions.
I would say that specificity in action declaration is desirable and, in the case of your example, "I try to push the statue using a technique my knowledge of these ancestral halls gives me" is a reasonably specific and succinct action declaration that is achievable by a player. As DM, I can now call for a Strength check and the player can add History. (Notably, players saying they are adding a skill proficiency bonus is a divergence from the rules which allow players to ask only. I skip that part in my games to reduce back and forth.)
The rules you appear to reference must be taken in the context of "How to Play" in which the player describes what they want to do before the DM calls for an ability check. The subsequent rules in the "Skills" section only say the player can ask if a proficiency applies to a check the DM already called for, but does not say the player can just add more description to make a skill relevant to the task. As with any other description by the player, that needs to be front-loaded. As I mention above, if the DM allows for it to be tacked on at the end, it's effectively a retcon and players would be well advised to always do this because it's to their advantage - start off vague, wait to see if DM calls for check, then add description to seek a bonus. Whether that's the game experience a given DM is going for is up to them, of course.