I always installed GNOME extensions via Firefox. You first install the GNOME shell integration Firefox addon after which extensions can be installed from the GNOME Extensions website. This always felt like an atypical way to install Linux software. TIL GNOME extensions can also be installed via the system’s package manager!
GNOME extensions can be installed via the package manager and are activated via the accompanying ‘gnome-extensions-app’.
Open the GNOME “Extensions” application to enable or disable installed extensions.
If I had read the notification at the top of the GNOME extensions website more carefully, I would have known this years ago. Oops.