You need the G account to be able to install apps from Play Store, I don’t believe the private space itself requires it.
Not sure if there’s some Play “integrity check” on stock ROMs, but on GOS I was able to create the private space and download&install F-droid or other APKs just fine, without a Google account.
I have to join the choir, what do you mean dying and doesn’t work? If proprietary apps don’t support it, it’s just because it’s one of the best ways to lock people in.