Firstly, just because something is Bayesian evidence, it doesn't follow that it's strong enough to overcome the prior probability.

Ah, that's the issue: I don't mean that it's more likely than not, or P(E|S)>P(~E|S), just that it's more likely than it would be otherwise, or P(E|S)>P(E)>P(E|~S).

I suspect you may be using a more general definition of "stereotype"

Quite possibly. What I mean by 'stereotype' is generally 'the general population noticing results from a distributional tendency.' Suppose the population holds an opinion of the form "men are smarter than women." As a logical statement, it is disproven by finding a single woman who is smarter than a single man (which is easy to do!). As a distributional statement, it could be interpreted as any of "the male intelligence mean is larger than the female intelligence mean" or "the male intelligence variance is larger than the female intelligence variance" or "high male intelligence is more visible than high female intelligence," because all of those are distributional tendencies that could have noticeable results along the lines of "men are smarter than women."

In particular, the ground truth of higher male variance in intelligence is interesting because it results in both "men are smarter than women" and "men are dumber than women" being valid impressions, in the sense that there are more smart men than smart women and dumb men than dumb women! This is perfectly natural if you think in distributions, and it seems to me that both of those are memes that are common in the wider culture.