Kagi already provides a way to search anonymously via a random email address (we do not really verify it or need it for anything) and Bitcoin/Lightning payment [1].
Since you are interested in cryptography, there is a discussion on Kagi feedback site along the same lines as your idea, about possible ways to achieve this without the need for cryptocurrency. [2]
Thanks for the links. Using a disposable email with crypto payments and occasionally generating a new account to unlink from previous searches could be a viable intermediate solution.
Also, I found this link [1] in the thread you mentioned. They seem to have implemented something like that.
Just to make it clear, Kagi does not link searches to an account already, to begin with. Refer to our privacy policy [1]. We simply do not need that data for anything and it would be just a liability for us. Our philosophy is that users should personalize the search feed themselves and this is why we built features like the aiblity to block or promote domains, create search lenses and many more.
However there is no technical way of proving it. So cryptocurrency and cryptography are ways to achieve anonimity from a perspective of a user, regardless of what we are doing.
Also, I found this link [1] in the thread you mentioned. They seem to have implemented something like that.
[1] https://metager.de/keys/help/anonymous-token