Okay, I've removed the extension, restarted, added the extension and enabled it to run in private tabs, then restarted again.
Searches are still happening on Google by default. A new search engine has been added as inactive.

I'm going to try making it the default search engine, which is an option in the dots menu.
Yep, now it works as expected again. So whatever bad state my Chrome profile got into, it seems removing everything and starting over does fix it, at least for now.