Its even stupider than that. There are only two major, publicly available web indexes in the USA, Google's and Bing's. After 24 February 2022, DuckDuckGo ended their partnership with Yandex, and since then they say "we have more traditional links and images in our search results too, which we largely source from Bing" https://help.duckduckgo.com/duckduckgo-help-pages/results/so...

The web indexes from Google and Bing are available publicly? I can pull it down from somewhere and try to make a search engine?

Bing at least license their indexes to partners on a commercial basis, as did Yahoo until they gave up indexing the web. I am sure that the NSA, the Chinese government, the ahrefs website, and other organizations have comprehensive indexes of the web which they don't share in this way.

Mojeek seems to be the independent, non-paywalled search engine with the biggest index, for an overview see https://seirdy.one/posts/2021/03/10/search-engines-with-own-...