They really don't, though Kagi costs $. Even ignoring that it's a bit unclear, since in many cases DDG or bing give better results.

Do you have any examples or types of query of where a free search engine produces better results?

I tried to Google for it, but maybe expectedly, that doesn’t bring up any good examples.

I think free has the be table-stakes for the default search engine, because otherwise you are forcing all users to change or buy a license, so any paid alternatives are out.

Disclaimer: Work at Google; not on search-related things, opinions my own.

Off the top of my head I can't offer examples, only my impression. In general Google seems better at narrowing in on what I'm looking for especially with more complicated queries, but in some cases the DDG results will turn up less spammy/more desirable results in the top few when I search with both.

I can only name one example where Bing gives better results, but it's only because Google intentionally doesn't: 123movies

It absolutely is a disagreement. I don't agree they have the best search engine at the moment. They did until they crippled it with the last 2 big updates.

Then who has the best search engine?

Free? Perplexity or Bing. Paid? Perplexity (pro).