Intelligence also seems to have plenty of culture-independent aspects.

This is partially true, but there are many significant markers of attractiveness that are not particularly context-dependent.

> Being judged intelligent is also a cultural thing that varies with time and place.

This I disagree with much more strongly. "Being judged intelligent" isn't explicitly isn't the only thing that feeds into the evolutionary drive, and the second-order consequences are much more significant: being quick-witted/funny, increasing likelihood of success/resources, etc. The concept of general intelligence is not nearly as narrowly-defined as you're implying. To take an example, subsistence farming isn't exactly _easy_. General intelligence is an important factor in success at subsistence farming just as it is in success at post-industrial white-collar work.

Also, while intelligence is per se attractive to some degree, it's also mediated through downstream effects like a higher likelihood of financial/resource success.