some do, even when homeschooled from the start.
all the parents I have known this happen to have sent them, though I personally can see the arguements for saying no. Think its a case by case thing, really. But I feel strongly about my kids having agency over their own education and so-if they want to go and it doesn't screw up family life for the rest of us-fine.
In all cases the kids wanting to go to school were teenagers. All bets are off really, then, IMO, anyway.
kids do often want to experience school, its a common cultural experience. So they go for a bit. And then they can always leave, its not a problem.
Thats the thing, its not a now-and-foreever decision. They can always go to school for a term or two then leave again.
The HE community has, IMO, a very strong identity and HEdded kids normally feel quite proud of this. Although they are statistically a minority, normally they will know other HE'd kids and so get an identity from that.