Vlad FWIW the Kagi extension was working on my iPad until this morning, when it started forcing me to log in every time I searched despite already being logged in (if I went directly to kagi.com I was able to make searches, but if I tried searching directly from the browser URL bar, I would get sent to a log-in page where I was unable to successfully log in). I just downloaded the test flight beta and that appears to have fixed the issue on my iPad.
System version: 17.4.1
Device type: iPad Pro 5 12.9"