I don't get the impression that Kagi intends to compete with major search engines. It is clearly marketed toward privacy-focused, tech-minded individuals. You can take that one of two ways. Either you are frustrated with the erosion of search engine quality due to advertising, or you disagree with the predatory practices such as data mining that comes along with such advertising. In both cases, the only real way to signal to major search engines that you disagree with these practices is to stop using their services (including their APIs).
For example, I have been using DuckDuckGo for decades. At first, I had to compromise search result quality, but now it has enough users and support that results are on-par with the likes of Google.
I do not think that Kagi is bad or that people should not use it. It simply isn't for me, because it does not actually address the reasons I do not use search engines like Google.