The latest update (2.2.4) broke redirections for me as well. Searches just go straight to the underlying search engine. (For the record, I'm not upset at you, Apple has made this complicated, I just wanna help get it fixed.)
I'm on iOS 16.7.7, the extension is enabled and both kagi.com and my search engine are set to "allow". I even re-installed the extension and restarted my phone, no luck. This is 100% reproducible for me, so I'm happy to help debug/test however I can.