I love the simplicity, but the names disappear once I close/reopen Firefox.

Simple and it works (unlike others). Just RC on the tab and select RENAME TAB and DONE !

Even in private window with nothing allowed, extension is greyed out and will do nothing. Zero results.

Sorry. Doesn't work for PDF tabs which was the only reason I wanted it. I like to keep tabs for manuals open in a session and they always only show a generic globe making them unidentifiable until opened. This cant rename those pages. Clicking rename doesn't open the pop up.

It does rename the tab and I like the type of pop up that it uses for renaming compared to other add-ons. However, it doesn't retain the changed name if the page happens to be refreshed. That's a significant problem for me. If that one issue were to be fixed, then this would be 5/5. :)
Edit: I've read that "add-on authors can provide a single response to a review". If the issue gets fix, lmk and I will change my rating.
Edit: Add-on now keeps name change even after refreshing! 5/5 😊