The problem I find with all pop-level proofs of Gödel's theorems and similar material, including this one, is that they gloss over a key component: how to make a machine that talks about itself. After the part quoted above, a blogger (not Smullyan) does go on to say:

The proof of Gödel's theorem shows that there are statements of pure arithmetic that essentially express NPRNPR; the trick is to find some way to express NPRNPR as a statement about arithmetic, and most of the technical details (and cleverness!) of Gödel's theorem are concerned with this trick.

No explanation of this essential part of the proof is given. Unless you do that part, there's nothing in the supposed proof to limit it to systems that include arithmetic.