So, I removed the old extension, I installed the new one (with 2.0.2 version) and did the configuration again, giving the right permissions to Kagi extension (as you can see from the screenshot), but I just did a search and I landed on DuckDuckGo website (which was my previous default).
I tested this from an iPad Air with latest (17.2) OS
P.S: I did read all the thread and I understand your intention was to make the Safari extension better, but unfortunately it’s not the case yet. Aside from not working as expected, it’s even more complicated to configure (with the engine in Kagi settings which must match the default engine in Safari settings etc..) but I understand from permissions perspective it should be an improvement. My only suggestion is to maybe test it (with opt-in in Test Flight) a bit more before releasing a new version, to avoid disruption