This is a very interesting idea that other search engines have tried before. Actually, the Brave search engine is built over Cliqz[6] that implemented this same idea but *without* the user's consent.
Copy pasting from an old comment I made about this "human web" crawler idea:
Both PeARS[1] and Cliqz[2] tried to do that. Both got direct support from Mozilla[3][4] but it looks like neither really kicked off.
PeARS was meant to be installed voluntarily by users who would then choose to share their indexes only to those they personally trusted, so the idea is very privacy conscious but also very hard to scale.
Cliqz, on the other hand, apparently tried to work around that issue by having their add-on bundled by default in some Firefox installations[5] which was obviously very controversial because of its privacy and user consent implications.
I still think the idea has potential, though, even if it's in a more limited scope.
[1] https://github.com/PeARSearch/PeARS-orchard
[2] https://cliqz.com/en/whycliqz/human-web
[3] https://blog.mozilla.org/press-uk/2016/06/22/mozilla-gives-3...
[4] https://blog.mozilla.org/press-uk/2016/08/23/mozilla-makes-s...
[5] https://www.zdnet.com/article/firefox-tests-cliqz-engine-whi...
[6] https://www.theregister.com/2021/03/03/brave_buys_a_search_e...