I got the impression that the FSF rather downplayed their own kernel development after Linux became available. I remember reading things like, you can still work on Hurd if you want to, but it's more of a research project.
I wouldn't say taking over (not anymore) but yeah the fact that it's basically an almost 100% "drop in" alternative to gcc for Linux makes GNU software a bit less relevant/central to Linux. For better and for worse I guess!
Oh if only it were that gracious. My impression was FSF spent some time scolding us that we should wait for Hurd (not to mention Guile). But the Hurd project got deeply bogged down and never got to usability. The closest to admitting reality was the time Stallman started saying we should talk about "GNU/Linux" and not "Linux".
TBH, Stallman's baby has always been the compiler suite and the development environment. As far as he's concerned, 99% of the moving parts are in the GNU part of the environment and people should acknowledge this by using the name.