I want to point out that I intended the whole scenario as entirely hypothetical. It does not really make sense for there being a limit on it, but at the same time...
- There are already things that do not make sense in the same way (spells, martial abilities like Action Surge, Superiority Dice, Second Wind, Hit Dice)
- You can also reframe it as, for example, mental focus or something like that. There are other systems that do similar things, like D&D 3.5, PF1 and PF2.
But D&D doesn't even really have rules for lock picking in the first place. This is part of the problem for me. D&D has vaguely defined skills that say things like "You can attempt to do X" but it says nothing about how long it takes or anything relevant. D&D doesn't attempt to simulate this aspect at all, since it has no rules for it.
Also note again that my example rogue has infinite uses of the basic lock picking skill, but has a few limited uses of "success with no risk of failure, instantly".
It's hard to say that my system is less realistic than whatever is already in D&D 5E (since there is basically nothing to compare my system to in the first place)