Disclaimer: Kagi user here.

Searches are not pay-per-use anymore. But you still have to be logged in to search. The premise is, that they don't store your searches. It is not their business model. You are the paying customer, not the ad firms. This is, ultimately, not verifiable. It comes down to some sort of trust. And I do trust in them. The developers are actually great guys, there is a discord where they answer immediately, and a discourse forum where you can submit bugs/features, etc.

They take a strong stance on freedom. They refused to implement a suicide prevention message, as they felt, that it wasbnot the job of the search engine to patronize the user.

There is such a thing as a 'session link'. You can get it from you account. With this, I don't think cookies are necesary. But the link expires so you have to do it probably every few days anew.

The thing is: I want to aupport them. They have cool features (up/downranking websites, GPT-4 access over their proxy, etc.) Is it better than self-hosting: no. But I don't wanna self-host. And search is something that costs. So any service that does not live off donations or some sort of payment, is suspicious.

I hope I explained some of my reasons. Please debunk me and make me cancel my subscription there if I am in the wrong. Im not a shill and actually interested in your concerns :)