I wonder if Private Space apps can be turned off the same way work apps can. If so, I could ditch Island and stop abusing the work profile as a way to implement a light battery saver.
It does. It's perhaps a bit less flexible but it's largely sufficient:
I'm using Shelter and I'm planning to switch to Private Space. The one thing that's a bit annoying at the moment is that third party launchers might not have UI integration with it. For example Lawnchair does not. So for now it might not be practical to use with those.
The biggest friction point for me is the fact that files can only cross the work profile boundary by using the Android's share sheet (or with cloud storage, I guess), and some apps (cough cough Meta crap) didn't like it when you shared a file they couldn't directly access with them. I didn't encounter any such issues recently though.
If you pause the work profile (there's a button in the launcher to do that), all apps in it get killed and their icons and widgets in the launcher get grayed out. If you tap a grayed out icon, you get a dialog asking you if you want to unpause work apps. I think there are ways to automate pausing, but I don't use anything like that and literally only have the pause/unpause button as a toggle for the intrusive apps.
I didn't read much yet, but does private space have support for a separate VPN connection? That's why I use work profile, I have it for my tailscal connection to my media servers back home when I'm away, but don't want to drop off my main VPN. So the work profile allows for this.