I must be missing something, then: if you take input password A and salt/hash it individually for each server, and the attacker finds the valid input password A

I must be missing something, then: if you take input password A and salt/hash it individually for each server, and the attacker finds the valid input password A via brute force on one, why wouldn't it work on all of them? They aren't brute-forcing to find the salt or the hash, they're brute forcing input password A.