The OP specified stereotypes that are not true. Here’s the relevant part of the OP:
If “almost all X are not like that” then obviously it isn’t true. Pointing out that some small percentage of X are like that serves no purpose I can see other than to try to build up the stereotype by claiming that there’s truth in it.
If what the person wants to say is “some minority percentage of Black people are really good basketball players, and so are some minority percentage of people of all other skin colors” – that’s true enough; and I’ll be astonished if I’m promptly jumped on by others on this board for having said so. Nor, I note, has anybody been jumped on in this thread for having agreed that currently, in the USA, Black people are overrepresented in professional basketball teams.
But apparently what some on this thread want to be able to say is “there is truth in this stereotype because look at this handful of people it does apply to!” Which is at best misleading. And I really can’t think of any reason to do that unless the intention is to leave the impression that this stereotype is accurate – and, if also saying that all stereotypes have truth in them, to apply that impression also to even nastier stereotypes.
– I haven’t, while I’m at it, heard of statements such as “men are on average taller than women” being called “stereotypes” even when people leave the “on average” out of the statement; except possibly when in the context of other stereotypes being combined with it, as in somebody saying, for instance, “the stereotypical woman is small, soft-voiced, and not very bright” or “don’t worry your little head about it”. But even if “men are (on average) taller than women” is to be considered a stereotype – that’s still a very long way from either “there’s a kernel of truth in all stereotypes” or "almost all X are not like that, but . . . "
And yet people do that all the time. Do you really believe that people who think “Black people aren’t as smart as white people” don’t make that assumption about most Black people who they interact with?
And if a stereotype doesn’t (improperly) serve that purpose: what purpose does it serve?
Cite, please? (Not in context of some specific case, but overall.)
So is what other people say about them. And what other people say about them is in many cases also subject to massive ignorance about them.