> There are no obvious ways to actually install scripts

Script installation works by loading URLs that end up on ".user.js". This can be done by clicking a link or manually entering the URL. You can also drag and drop files ending up on ".user.js" to Firefox. Finally you can use the "Utilities" tab, enter a URL at the "URL" section and click "Import".

> The support site doesn't quite work (has a frozen pop-up on top)

Do you have JavaScript blocked/disabled i.e. via NoScript?

> but what shows gives Chrome specific instructions

This is on my TODO list. However, many answers apply regardless of browser in use.

Google Oauth is done via client-side authentication, which needs a redirect URI to pass the access token on success. The token is part of the URL's hash component which is _not_ transferred to the server. In normal Tampermonkey should close the tab after it extracted the access token from the tab's URL.

https://developers.google.com/identity/protocols/OAuth2UserAgent#handlingresponse