But there being less full professors than assistant professors isn't evidence of them not being promoted on the basis of their sex? Are they dropping out at different rates or different ages? Are they putting in the same hours? Are they accomplishing the same things? Are they in the same fields? The analysis of the gender earnings gap showed that these assumptions that all of these things were a given to have been equal between the sex populations were actually far from it. There are so many variables at play. The argument would be much more compelling if it didn't just offer the single variable of their sex, with the assumption being that all else considered, the candidates are identical.

Well, whether or not IQ correlates to intelligence, we do know that males tend to have greater variance in IQ while females tend to be more clustered around the mean - while also having a higher mean IQ.

If IQ correlates to academic success (which I believe studies have shown it does) then this would lead one to reasonably conclude there should be more female undergraduates and graduate students, and an equal number (or possibly slightly more) male assistant professors; we'd also expect a greater proportion of full professors who are male. This is in fact what is observed.

I'm not a biological determinist, and it's certainly possible the IQ test itself is sexist/ not unbiased, but this is one possible explanation where we can observe what we do without the presence of sexism.

It's also certainly possible there IS sexism preventing women from becoming professsors, I'd just need to see more compelling evidence than what's presented here.

This argument would only make sense if hiring for academic positions included random samples.

No, not really; the applicant pool is, by definition, not random.