Doesn't show where? In the Extensions tab or in your Safari toolbar? Can you provide some screenshots?

Have you used the Safari extension on this computer before?

In any case, I'm afraid there's no other answer than what's on the documentation page. The plugin gets registered by macOS, not by us, and if it's not working, that's a macOS bug.

Note that, unless you're running a Mac from 2015–2017 or earlier, you should be able to upgrade to a more recent version of macOS. If you can't, you should at least upgrade to the latest version of Monterey.

Beyond that, delete the app from Applications, restart your Mac, redownload the app, and open the app. If the plugin doesn't appear in Safari after that, you may just need to use another browser with a more reliable extension framework. But this can almost always be fixed by following these steps precisely.

@NicolaPhilbin: There's no "lack of concern" — we just can't fix something that's not our bug. The fact that you can compress the app, delete it, and uncompress the exact same app package makes 100% clear that there's nothing wrong with the app itself. Safari's proprietary extension framework is just vastly buggier than other browsers' frameworks, and has been for years.

But as I say above, you can almost always fix this by 1) upgrading to a current version of macOS and 2) precisely following the given steps. If that's not working, there's something wrong with your macOS installation.

UPDATE:

I updated to the newest version of Monterey, then I deleted Zotero, then restarted my computer and downloaded Zotero and then opened Safari after that. The Zotero symbol returned to my Safari Toolbar!! @NicolaPhilbin try it, maybe it works for you too!