“Kagi it” feels clunky to me. “Google it” while I’ve heard many instances of people using the phrase even when Google isn’t involved has a ring to it that’s easy to say and tools out well.
Have you given thought to whet the Kagi equivalent would be?
One could also argue that google is a pretty nice word for the English language since it evokes words like "look", "oggle", "oodles", implying looking through a lot of stuff and having a playful tone. Sounds silly but when someone says words sound like they mean something, it is a real phenomenon due to the way we interpret various sounds. In most languages, B sounds sound like big and round things and K sounds sound like sharp things. There have been studies on this. You can then get more specific in the subconscious patterns people recognise when you narrow it down to a certain language. This is why I say, Google is a great word for its function in the English language.
I don’t think it really matters as long as meaning is there.
Though ‘Google’ is starting to mean ‘ad infested trash’, so maybe the word will lose favour.