nompan I understand it is frustrating, but let's try to move us forward.
Please share your phone, ios version as well as queries or video to reproduce the problem. On my phone it seems to work fine.
Note, this is only testing it anectdotally as I am using Orion browser day-to-day which has Kagi built it.
Also a reminder, that a core issue of the problem is Safari being only browser where you can not chose your search engine. Submitting product feedback to Apple and asking to allow users to select any search engine is something we should all do.