Kagi currently requires an account/email to sign up.
Search is a very intimate service, and I am very hesitant to use a service where it has full knowledge of my searches because it's linked to an account/email.
An anonymous email can be used, but paying for the subscription with a credit card requires giving personal information. A solution is to enable payments through cryptocurrency like Bitcoin or Monero.
Mullvad, the VPN service, is a potential model to follow. In order to use their service, you send a payment which instantly generates a code, and then you input the code on their app. There is no account or email needed and payment can be done through a variety of ways including credit card, Bitcoin, cash in an envelope, etc.
The privacy policy states that no personal information is collected, and while that is great to see, a much stronger guarantee is to enable anonymous payments.