I think you are not a computer programmer. Trying to build an index of the web by querying other search engines is not an efficient or sensible way to do things. Using ActivityPub for it is insane. Sharing query results in the obvious way might help a little during events where everyone searches for the same thing all at once, but in a relatively small pool of relatively sophisticated Internet users I don't think that happens often enough to justify the enormous amount of work and complexity.
On the other hand a distributed web crawler that puts its results in a free and decentralized database (one appropriate to the task; not blockchain) might be interesting. If the load on each node could be made light enough and the software simple enough that millions of people could run it at home, maybe it could be one way to build a new search engine. If that needs doing and someone has several hundred hours of free time to get it started.