They are constantly telling all their GHES customers who complain about the severe flaws with the self-hosted appliance product to move to GitHub Enterprise Cloud, which is just regular GitHub.com, but who in their right mind would make that move nowadays??? At least GHES stays up during the daily github.com outages.
It's still a pretty annoying process, though.
I wish they had a plan to literally host GHES for you because then more people in the company would be forced to reckon with how terrible GHES is from an operational perspective. It is stuck ca. 15-20 years ago conceptually.
They don't host github enterprise server for you (though gitlab has something called gitlab dedicated which they host gitlab ee for you).
Perhaps this header mentioned in the article is related, maybe that's the toggle for the enterprise mode? Seems there is at least traces of "enterprise mode" on the normal github servers.
Any public instance should update immediately though, it's not very hard to put together how to repro the vulnerability on your own from what they provide in the article and the fact that GitHub Enterprise source is publicly available.
Guess which is usually picked ...