They should make a search engine. If Kagi can do it, why can’t Mozilla?
The biggest provider of Kagi’s results is Google. They are unique in that they have their own Tinygem and Teclis indexes to augment results, though. Mozilla could certainly operate a plain Google proxy like Mullvad does with Leta, but I don’t think they’d be making more money out of it than just agreeing to Google’s exclusive terms.
Building a search engine with an independent index is hard. Mojeek has done the best job of it, but you can tell there’s a disparity in result quality even if they’re improving.