Well put, especially because all this is really about is averages.
The GP's post title is interesting - he refers to the ex-president of Harvard who lost his job (in part) due to comments - that were pretty completely misrepresented - about the aptitude of women in mathematics and science. His basic point - which isn't particularly controversial - is that men tend to have a greater standard deviation for many characteristics than women do, although they tend to have approximately the same averages. This wider bell curve would obviously lead to more men at the highest levels - but also at the lowest. Both of these are well supported observations (especially the low half, which is less controversial - go figure) and are also reasonable conclusions from a genetic standpoint - women have two X chromosomes, which moderate each other, while men have only one (this is why e.g., color blindness are almost exclusively male afflictions) - as well as an evolutionary standpoint (a population-exceptional male can have dramatically more successful offspring than an average male, but a population-exceptional female will have approximately the same number, although more fit, as an average female).
Stereotypes are all about averages, and the reason they've so pervasive is that they're how our brains work. We need to have a mental model of everything around us, so we don't spend 10 minutes trying to understand an apple every time we see one. We couldn't function if we weren't able to say "this is an apple, it acts like the other apples I've experienced" and put it in that bucket.
The problem doesn't even arise when we do that with people - to an extent. We have a stereotype of doctors as intelligent and knowledgeable about our health, for instance, that's usually quite helpful if we're a patient. No, the problem comes when we don't remember that stereotypes are just personal averages, and that a specific individual may not fit the model we have, combined with trying too hard to fit people into buckets when the evidence doesn't fit. The other problem is not discarding a bucket when essentially nobody fits it (e.g., common racism, sexism, anti-Semitism, etc - all of which persist only because the bigot in question has spent their life cherry-picking and exaggerating interactions).
(Also, back on topic, women aren't any worse at spatial reasoning, they just - on average - take slightly longer to do it)