It is in the AUR, but be mindful to grab the git version--the original project is no longer maintained (the link you posted is from 2020).
paru nautilus-admin-git
I just tested it and it works.
The problem with adding this or any other extension to the Gnome spin is that Gnome has a reputation for destroying extensions when they release version updates. The Gnome team is very open about the fact that extensions are not supported, and they cannot be bothered to give a crap about what may or may not happen to extensions when they make sweeping back-end changes.
Frozen pool distros like Ubuntu generally have an easier time keeping extensions like this alive, but still it can be a lot of work for maintainers to stay on top of breaking changes. For a rolling release distro, the nuisance is more persistent and fast-paced.
All that to say: I wouldn't hold your breath for the package being added to the official spin.
If you try out the extension and you like it, you could ask very nicely in this thread and the guys might be willing to add it into the Chaotic AUR so you don't have to keep up with building it yourself.