You can sign up to Kagi and pay anonymously, they state how to do it in their FAQ's.

I was thinking about the aspect of privacy. Not only do you need a user, you also have to give them your payment details.

Kagi has established Privacy Pass which perfectly meets the need of balancing paid search with privacy. If you have a better fix I'm sure they'd love to hear about it on their feedback forum.

And don’t get me started on their “300 searches a month”-bullshit. If I pay for 300 searches, I want my 300 searches. Regardless of how long it takes me to use it.

¯\_(ツ)_/¯

As a someone who averages >1000 searches/month, I burned through the 300/month tier pretty quickly. They're kind enough to credit you if you don't use their service in a given month, which is already uncommon in the subscription business model.