Assuming you mean a negative stereotype, so for example not just a basketball player but one who does it instead of studying academic subjects: then certainly, yes. Imagine someone saying "oh those guys won't have a clue, they're just a bunch of Shaqs". Even if Shaq is the name of a real person, the racist undertone is clear.
> ... so is that ... sexist?
Not so much, maybe borderline, just as Karen is only borderline racist.
> The original Karen (from /r/FuckYouKaren) was a real person named Karen.
I dispute that many people using the term know exactly who that Karen is, and even moreso the people hearing it. To most, it simply sounds like a generic name that white middle-class women are likely to have.
(After a little Google research it seems there actually isn't an original Karen. KnowYourMeme (not exactly a reliable publication but relevant here!) had a few candidates but no clear winner - in fact the main contender was just a bad comedian using it for an unspecific annoying woman. In a post I found on /r/FuckYouKaren itself no one had any ideas. But that's all irrelevant anyway, if people using the term don't know.)
> If using a woman's name is sexist then so would using a man's name. ... And if it's "fantastic" to select a female ... sometimes it goes the other way too.
Actually, by "avoidable" I meant there's no reason to use any name at all. But let's put that aside.
Look, I get it, I honestly do. Equality really means treating people equally, and that means if we're allowed to use a male name as an insult then we ought to be able to use a female one too, right? I actually do think it makes a lot of logical sense.
The problem is that it only makes sense if the current base line is men and women already being treated equally - then adding extra opportunities, compliments and even insults equally would preserve equality. In practice, current societal norms are absolutely flooded with little microagressions that subconsciously make men and even women themselves think of women as somehow less important than men. I remember a very awkward encounter at a team building exercise where a colleague (actually a fantastically nice guy) jokingly let me go first at something with a cheeky "ladies first" comment, only to realise a female colleague was actually nearby (near enough to cause a bit of confusion!). Note that one woman jokingly accusing another of being a man would never be an insult, even jokingly.
Yes I realise there's the odd extreme feminist that genuinely thinks men are less important or worthwhile than women (e.g. those that say that men accused of sexual assault by women should be assumed guilty). But those numbers are absolutely dwarfed by men who think less of women. In that context, your idea (allowing insults that reinforce stereotypes against women in the name of equality) does not make sense.
Suppose I use "Shaq" as a stand in for a generic basketball player. I'm obviously making reference to a specific famous basketball player who happens to be a black man, so is that racist and sexist? Asians can play basketball. Women can play basketball.
It's not chosen at random, it's chosen as an exemplar of the category. The original Karen (from /r/FuckYouKaren) was a real person named Karen.
> historically, it is sadly a stereotype that women's opinions are more petty and unimportant than men's. The use of the word "Karen" for someone with unimportant, ill-informed opinions is directly playing into an existing stereotype. And it is totally avoidable.
I don't think it is avoidable. If using a woman's name is sexist then so would using a man's name. Either way it would cause the reader to be more likely to picture the person with the contemptuous attitude as that gender. But so would not specifying one at all, which we know generally causes people to picture a male. So you have to make a choice one way or the other.
And if it's "fantastic" to select a female in the contexts where the subject is a meritorious student or engineer, sometimes it goes the other way too.