I regularly use a VPN and/or incognito mode. I am not logged into Kagi there, which means that every time I use the browser's navigation bar to search something (without explicitly specifying another search engine), I land on the login page for Kagi as it is my standard search engine. Same goes for my VPN sometimes, where it seems to have been blocked (at least on some IPs), as I am getting 403s when visiting Kagi.
Therefore I would find it incredibly helpful for search queries to be automatically redirected to another search engine when the user is not logged in.