But if you chose to incorporate and run a company in the US as YC requires, and participate in their program, you'd be "public" regardless, in various forms.

There's being in public, and there's being put on a pedestal (:

As far as I can see, choosing to become a founder in YC and succeeding at it requires a bit of desire for the limelight.

Not necessarily. With young startups, the only available position for a role is often "Founder". A developer could simply want to work on a project and see where it goes, and the only way would be to become a founder.

In general yes. But with YC, I get the feeling that there is a push for founders who at the very least won’t actively shun the limelight if required.

Yep: There's a reason a bunch of YC founders appear every year like clockwork on popularity lists like Forbes 30u30 ...