I've had the same thoughts on those. Maybe it also allows them to gather data on whether/how people interact with a webpage itself, which may be an indicator for its quality.

Yes! It tells them what people search for on that particular website… all very valuable data.

Hmm I’d like to hear more from Kagi on this. They make a point of not logging searches by their subscribers. If this widget is available to non-subscribers outside of Kagi.com, do the same privacy-first principles apply?