Sorry I didn't get back sooner, but I made some progress.

What do you mean with "work in progress"?

Their words (second video, I think), and more in reference to how they are still working out how they haven't yet covered all of the use cases (like maybe my needs can't currently be met by rpm-ostree or bootc). rpm-ostree has functional limitations, and bootc is still being developed. Obviously, both are still useable and useful, and Universal Blue has been using them for quite a while. I may have been reading too much into it with the "depreciation" comment.

So, did you try the following methods when installing the .run file? If so, how did it go?

It can't work on its own. Running with sh or making it executable runs the script, but it fails when it tries to write its icon and .desktop entry to /usr (it also doesn't take an --appimage-extract argument). You can use sudo rpm-ostree usroverlay to create a temporary FS overlay for /usr, but it's wiped on the next boot. Still, that allowed the installation to complete.

I discovered that it's installing all of the necessary components to /opt, and they remain functional. I was able to manually run the daemon script required and get a WireGuard tunnel established in the client.

Now, I'm trying to get a .service module to work so it can run automatically as root on a reboot with systemd. So far, it's giving me a 126 exit code, so I still haven't figured out how to escalate its privileges automatically, but this is the most progress I've made to date.