There are services that allow anonymous payments. I think Proton, Mailbox.org, and Posteo all accept cash payments.

Kagi has a “privacy pass” feature, which uses some fancy ephemeral authentication tokens that are not traceable to your specific account. You lose personalized settings, naturally (like site rankings) but you can do searches that aren’t directly tied to your account.

For email, there’s no one-size-fits-all risk model. I mean, my email address is my full name, @ a domain that I own and is itself traceable to me. I have no need for anonymity, but I DO need privacy and security.

Now stand in awe of my wild, wild rolling!