In worrying about whether the verifier "uses" the certificate, you're taking the term "verifier" too literally. A verifier is an algorithm that takes the original problem and some additional information ("a certificate"), and provides the correct answer ("yes" or "no") in polynomial time. We call it a verifier, but that doesn't impute upon it a new "job".
For problems like subset sum, the certificate serves as a useful shortcut- we're given a subset that we just have to check a) adds up to 0 and b) is a subset. But if we already know the problem can be solved in polynomial time, that shortcut becomes unnecessary.