I think you have a fundamental misunderstanding of Bitcoin, rather than how pool works. PoW is not a random number, PoW in Bitcoin is defined with the hash that is difficult to find, ie. your leading zeros in simplified terms.
No, I understand that. It's a terminology problem. I've been using PoW to refer to the hash whether it is valid or not. I see now that PoW is defined as only the hash that has the number of leading zero bits.
I'm not sure I understand the difference between leading zeroes and difficulty though. People throw around the term difficulty rather loosely as though it is a euphemism. I prefer concrete terms.