When I select one, they stay greyed out.
When I said "purged", I also deleted all the mozilla profiles and started with clean installs.

blinkyd, I'm sorry you're still having problems with Firefox. Out of interest, are you using the Mint-prepared package of Firefox from their APT repository or are you using another version (e.g., a Flatpak)?

You could try downloading Firefox directly from Mozilla and installing it in your home directory to see if it has the same issues. It's far from ideal, but it could at least point to whether the issue you're having is something that the Mint team might have done inadvertently when packaging (assuming you installed it from their repository) - even if it only shows up in niche circumstances. And if it works as expected, it could be used as a workaround until the issue is corrected.

BTW, what is "Brave search"? Is it just a search engine, or some sort of addon?

I have to say, OP is not very forthcoming with detail here. Just venting I guess.

There's a questionable add-on for the search engine, but I added it to my search engines list this way.

EDIT:

Whilst I think of it, another thing the OP could try is enabling Troubleshoot Mode (Help > Troubleshoot Mode...) and see if that enables options to reset his search engines.

If that doesn't work, whilst in Troubleshoot Mode enter in your address bar and click Open Directory for Profile Directory (it'll be in ~/.mozilla/firefox) and, keeping that open in your file manager, return to Firefox and quit with CTRL+Q. Let Firefox clean up and then in your file manager window find search.json.mozlz4 and move it to your home directory or wherever. Restart Firefox and see if it rebuilds your search engine list.