It would be great if the URL bar showed the phrase searched for, as it does for default search engines, rather than the kagi.com url in the search bar.
bs I don't think this is possible (Safari limitation). Can you check if other search extensions are able to do this (xSearch comes to mind)
bs Not possible (Safari limitation). Use Orion browser with Kagi built in 🙂
xsearch doesn't do it either. I think as Vlad said, it's a safari limitation only allowing to do it with their own search engines