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]

[1] https://blog.kagi.com/accepting-paypal-bitcoin

[2] https://kagifeedback.org/d/653-completely-anonymous-searches...