Vlad this is really cool and I think it shows Kagi is a company that's serious about its privacy marketing; so both thank you and congratulations!
The biggest drawback for me is definitely the loss of search prioritization. I wonder if search prioritization options could be encrypted with the login password (kind of like a password vault). The client would then decrypt the "settings vault" and pass along the settings to the server.
This would allow searches to be tracked, however, the Kagi servers (under this model) would not be able to match the search settings back to the user ... because Kagi doesn't actually know who owns them anymore.
Barring that, I think what I'd like to see is the extension have an option to detect when its in a private browser window and automatically enable itself only for private windows (so I get some automatic control over when privacy pass is used based on how I normally use my browser).
Also, it seems the Privacy Pass option does not yet work for Firefox for Android (it says it's not compatible with Firefox mobile), so that's a bummer.