Not sure what you mean by "private payments" but Kagi accepts Bitcoin: https://help.kagi.com/kagi/plans/payment-methods.html

Monero ideally combined with something like this but one or the other would also be an improvement:

>Brave uses a blind token protocol based on Privacy Pass, which decouples payment identity from service usage. These privacy-preserving subscription credentials allow the browser to verify you have a valid purchase of a premium product (like Origin) without learning anything about you.

https://brave.com/origin/