My claim is that "capturing [...] some pervasive intuitions" is not what ZFC is trying to do, or what people in foundations are interested in. Lurie puts the point well when he says a foundation "mean[s] a small number of concepts and assumptions from which the rest of mathematics can be derived." The purposes here are first, to figure out what exactly one must assume to get the rest of mainstream mathematics, and second, to provide a convenient base for doing metamathematics (since the fewer elementary operations/objects you have to reason with, the easier it is to write proofs). Capturing intuitions is completely orthogonal to this endeavor; no one is claiming that set-theoretic translations of everyday mathematical arguments capture the intuitive content of those arguments, because no one one intended ZFC to ever do that in the first place.

Similarly, no one is claiming that handing someone a Turing machine that performs quicksort is a great way to capture the intuition behind quicksort, or explain the algorithm in an understandable way. Turing machines exist as a special-purpose construct to probe the limits of computation, not to reproduce or communicate high-level algoritmhic reasoning. ZFC is like a Turing machine for (meta)math.