Vlad astronaut
I made sure the extension permissions were set up correctly, they were set to "allow" for both Kagi and Other Websites already. I have since updated to iOS 17.5.1, still no dice. It redirects to the login page instead of search results when I search (even though I am logged in), and curiously enough, logging in through that page anyway gives me a 403. Logging out and back in, clearing cookies, reinstalling the extension, and trying the testflight version make no difference.
My current workaround is going to Kagi.com and searching from there, as unfortunately using Orion isn't a suitable replacement for me at the time (Nothing wrong with the browser, I'd love to use it - I just rely on a couple of extensions that don't work). Sorry Apple switched things up by surprise on you all - hopefully this is figured out soon.
EDIT: Strangely enough, going to the google homepage and searching from there does successfully redirect to Kagi, it's just that doing it thru the search bar doesn't work. I don't think this behavior is necessarily tied to an iOS update though, because I was seeing the same login page behavior before I updated from 17.4.