I use jabref and this extension quite heavily. I can assure you that it does send the URL to jabref; it gets added as a Misc reference with the site URL in the optional fields. On my firefox / windows system it does show greyed out in the plugins menu like you say, however it adds a jabref logo in the address bar which can be clicked (or alt+shift+j) to send to jabref.

I just tried it on my linux system though, and it doesn’t work for me, either. Suspect some sandboxing weirdness because I have jabref as a flatpak but firefox running natively. I’m just coming back to linux from a few years hiatus so I’m hoping someone better than me at this can check in.

Jabref does have some troubleshooting steps for their extension that might be worth trying though, depending on your install.