You also have no idea what kagi is doing with your data

We in fact do have an idea what they do and don't do:

• Searches are anonymous and private to you. Kagi does not log and associate searches with an account.
• We do not log or store your IP address. Your IP address is used only temporarily when enriching location/maps searches, and is not shared with any other party.
• We only store cookies needed for site functionality.
• We do not use any web browser analytics or other frontend telemetry.
• We do not display any ads, or have any first-party or third-party tracking in service of ads.
• We do not share customer data with third parties, except as needed to perform explicitly accessed services. In those cases, we will share the minimum amount of data needed to provide the service, and will do so in an anonymous way.
• We collect only the data needed to provide and protect the service.
• We proxy all images to prevent tracking from third parties.
• We use HTTPS encryption everywhere. All passwords are hashed and salted.

https://kagi.com/privacy

These terms are legally binding. If they did log searches despite these terms, that could end their business.

it’s inherently eventually unprivate since it relies on a login.

Not anonymous != unprivate.

Even if it was, I don't think it's different for all of the other search engines. For example: I do not believe for one second that Google can't identify you without being logged into your account; even with all the blocklisting your typical ad-blocker does.
Go try and fool https://abrahamjuliot.github.io/creepjs/ if you want to go try how little even things like incognito mode help against identification on the web and this is all just relatively simple client-side analysis without behaviour tracking.

There is nothing wrong with ads

I disagree that there is nothing wrong with modern propaganda but that's a topic for another discussion.

The search results on free search engines are also the product here, since they only get paid from using them for results.

No. That's the thing, they're not. Search results only serve to attract users. They only need to be good enough to be acceptable to users; everything beyond that is a waste of time and money from a business perspective.
They receive exactly $0 from you as a user. There is no sale contract between you. Therefore, you are not their customer, you are the product they sell to their actual customers.