Well, I don't think it's safe to say natural numbers "are" sets, but surely they are isomorphic to some collection of sets (and this allows them to be modeled a

Well, I don't think it's safe to say natural numbers "are" sets, but surely they are isomorphic to some collection of sets (and this allows them to be modeled as sets within set theory). The important part about the construction of the natural numbers from axiomatic set theory is that it can be done, not that it brings us closer to the Platonic idea of numbers. It can of course be done in many ways (OP's post lists just two). There's no reason to believe any specific representation within set theory is the true order of the universe, but it is extremely useful and we should be glad it works so well.