The most helpful comment caught a nasty thinko in one of the proofs. But the most interesting observation this referee made was that perhaps the central argument of the paper is a so-called "squeezing argument". (See Robbie Williams's discussion of such arguments here, and Peter Smith's here, a version of which was eventually published in Analysis.)
The rough structure of such arguments is as follows. Suppose there is some intuitive notion I and you want to show that some rigorous notion R is co-extensive with I. Then one way to do so is as follows. Suppose that it is uncontroversial that R gives a necessary condition for I. And suppose further that we can find a different rigorous notion Q that uncontroversially gives a sufficient condition for I. So, to put it set-theoretically, we have:
Q ⊆ I ⊆ RThen if we can show rigorously that R is sufficient for Q, i.e., that R ⊆ Q, then it will follow that both Q and R are co-extensive with I. As it's put, I has been "squeezed" between Q and R.
The way this works in the paper is that I is the intutive notion of the ancestral; R is Frege's definition; and Q is an alternative definition that I give and claim, in fact, is intensionally correct. In response to an objection to the intensional correctness of that definition, however, I fall back on this squeezing argument.
This makes at least three instances of this sort of argument: The original, in Kreisel, which is meant to show that the model-theoretic account of validity is extensionally correct; Smith's, which is supposed to show that Turing's analysis of computability is extensionally correct; and now this one. Are there others? I'm guessing maybe there are?