If you're given the algorithm to begin with, this doesn't seem like a big issue. But if I give you some code, which is an implementation of bubble sort but not labeled as such, then proving it takes quadratic time is more involved, and may (will?) require running the code. Right?

I'd say it's easier to prove that a naïve bubble sort runs in quadratic time, than it is to prove that when it terminates, the list is actually sorted. It's obvious by inspection that it contains a nested loop over the same data structure.