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I'm not 100% sure this perspective works, but could one take the position that this is because of a disconnect between model and reality?

You look at some big list of symbols and say "this formula (model) corresponds to whether a Turing machine enumerating all theorems of ZFC produces 0=1 (reality)". But does it really?

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In a way. Consider a Turing machine that enumerates all proofs in ZFC and halts iff it finds a proof of 0=1 (i.e. a contradiction). The statement that this Turing machine halts (like the statement that any given Turing machine halts) is what’s called a Sigma_1 statement, which lies at the bottom of the arithmetic hierarchy (right above bounded-quantifier statements). ZFC + ~Con(ZFC) is consistent but Sigma_1-unsound and therefore omega-inconsistent [1].

[1] https://en.wikipedia.org/wiki/Ω-consistent_theory

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