A search index isn’t a public facing search engine or service.
They’re building a common database for them to both use, as well as others.
So trying to beat Brave or any other search provider, isn’t the goal of this project.
It’s about creating a new index, Brave, or Bing, or other search providers could use if they wanted.
They also suggest splitting off Android and sharing data. But that seems barely a start. If you’re going through the effort to take them down, take them all the way down.
They should split off nearly every major product and service. Make it a couple dozen separate companies.