Could you please clarify your question here?

Why, as you come to believe that Zermelo-Fraenkel set theory has a model, do you come to believe that physical time will never show you a moment when a machine checking for ZF-inconsistency proofs halts?

I try to intepret it (granted, I interpret it in my worldview which is different) and I cannot see the question here.

I am not 100% sure whether even PA has a model, but I find it likely that even ZFC has. But if I say that ZFC has a model, it means that this is a model where formula parts are numbered by the natural numbers derived from my notion of subsequent moments of time.