So everyone in the human-superiority crowd gloating about how they're superior to mere machines and formal systems, because they can see that Godel's Statement is true just by their sacred and mysterious mathematical intuition... "...Is actually committing a horrendous logical fallacy [...] though there's a less stupid version of the same argument which invokes second-order logic."

So... not everyone. In Godel, Escher, Bach, Hofstadter presents the second-order explanation of Godel's Incompleteness Theorem, and then goes on to discuss the "human-superiority" crowd. Granted, he doesn't give it much weight - but for reasons that have nothing to do with first- versus second-order logic.

Don't bash a camp just because some of their arguments are bad. Bash them because their strongest argument is bad, or shut up.

(To avoid misunderstanding: I think said camp is in fact wrong.)