While this is certainly true, I’d like to highlight the difference between privacy and anonymity. Kagi is worse because they require payment information (though there is an option which attempts to separate your payment from your searches), but they are better because their privacy policy is better. If you don’t trust the service itself, whether because of its location or any other reason, then you need to be paying in cash or monero, using free services. In both of those cases though, you also need to ensure you don’t deanonymize yourself in other ways. In other words, pay if you don’t want to think about too much, and you more or less trust the company.

I believe that even if it were GPL licensed you could do that, since you’re only running it on your server, not publishing it. Of course, for this very reason, most copylefted server software uses the AGPL, which does require source code of a network service to be made available.

While this is true, it’s also worth noting that the original is permissively licensed. You can change it if you want, it’s just that most people don’t bother. Presumably changing the image would be as simple as replacing the three files in /web/static/img, but maybe that would look weird for some reason.