Not an answer but better alternative: Kagi. Paid, but it pays for itself. Not affiliated. Just satisfied.
How about for people where having to log in to search is a nonstarter? Has a method to be able to take payments without the ability to track the users been invented yet?
If you are not comfortable with trusting their claims of not saving search history(which is very reasonable not to belive so imo), you don't have to use it. They also take Cryoto so if you think crypto is .. a thing, you can do that.
Use a burner email and pay via Bitcoin?
Kagi does not have an equivalent to Bing Chat does it. The latter is an AI.