Shor's algorithm and offset elimination
Yes, if you could consistently eliminate the offset then that would be enough. But how are you going to do that?

The state's offset is the discrete logarithm of the value in the ancilla register. But you don't know how to compute the discrete logarithm efficiently classically. And the offset changes from sample to sample. And there are so many possible offsets (assuming you picked a big number to factor) that you won't see the same one twice.