ZFC is the universally, unequivocally best definition of a set

Worse. You are being tricked into believing that ZFC is at all a definition of a set at all, while it is just a set of restrictions on what we would tolerate.

In some sense, if you believe that there is only one second-order model of natural numbers, you have to make decisions what are the properties of natural numbers that you can range over; as Cohen has taught us, this involves making a lot of set-theoretical decisions with continuum hypothesis being only one of them.