Extra PT is great for simple bullshit (on spot corrections, ect), but when there is an actual shortfall in training you need to correct the problem. You do that by identifying the problem; developing a retraining (or corrective) program (with some suck mixed in) and you implement it until the correction is made. Pushups will not fix falling asleep on guard duty, pushups doesn’t change the fact that you had a security breach and now have to report it, pushups does nothing for this situation at all.
When a soldier puts not only “you” but everyone who is relying on him for security at risk, it’s not leadership development time, it is get security up, report, check, confirm and check again time. Afterwards, if you are given the opportunity to correct the problem with your soldier, by all means do what you think is right. I can tell you from personal experience that pushups are not going to fix it, but run your soldiers how you see fit…