Even if it was open source, how would they prove anonymity?

I dont remember exactly, but it's possible to create a completely anon search engine. User encrypts the query using their key and sends it to search engine, search engine now doesn't know the query, but then it encrypts its entire database using public key of user and see if hashes meet for keywords?... something like that

I think you might be referring to homomorphic encryption. If I recall, the issues with it are the operations are expensive and it has a large overhead.

Also - how are you going to block bots without being anti-user also without some sort of tracking?

You could easily get 100 not requests for every 1 user request.

Our search server receives the request from the web server and returns the results. The web server, which could store the user IP, does not know the search query, since the POST method is used for transmission. The search server that processes the request does not know the user IP, only the web server.

The logs will be deleted after the visitor numbers have been determined. Your search query remains anonymous at https://tuxdex.com.

Easy: just don't block bots.

Problem solved.