They used to, but the plugin was removed in 3.36.
Trying to deal with all kinds of installable-things-that-aren’t-apps turned out to be quite a strain on the code base, so the developers decided to focus on the main use case.
I must say that it was quite the bummer to find out just how incompatible the whole browser integration is with sandboxing. It looks like we’ll have to bite the bullet and think about implementing installation support in the extensions app …
Except that for showing the actual extension preferences, tweak tool always launched a small helper program provided by gnome-shell. The “new” Extensions app isn’t actually that new, it’s an improved version of said helper program.
If it makes you feel better, think of it as cutting out a middle man.
(There’s also the tiny little detail that the app is actively maintained, while tweak tool has not seen a single code change since 3.34.0)
Almost all is a stretch IMHO, but some of the options are indeed being integrated into Settings.
and dangerous. Don’t get me wrong, live-patching the compositor is powerful, but it can never be as safe as - say - chosing between AM-PM and 24-hour clock format. Nothing in Settings should be able to break your entire session, which means that extension management will be kept separate.