Thinking about the geometry of that, it seems to me that any two things that correlate positively with all of a "vast battery of non-IQ-tests" are likely to correlate positively with each other, and the more diverse (i.e. less correlated with each other) that battery is, the tighter the constraint it places on the set of things that correlate with all of them.

Indeed. To the extent that existing IQ tests are not terrible & crappy (and, y'know, they aren't), any better IQ test is still going to correlate heavily with the old ones.