Lol Federation is the definition of scalable. Everyone serves their local users -> a miniscule amount of global traffic, everything but auth always stays local.
Universities have been doing it since the beginning of the internet. Email is the biggest example but there are others: eduGAIN and eduroam are the most notable ones coming out of the academic community.
No search engine has a "single point of entry". Every search engine has Cache servers all over the world at almost every major IXP. Nothing would prevent a federated service from operating the same way. Cloudflare or literally any form of loadbalancer or load balancing service could be used to redirect queries to fedisearch (or whatever the service name would be) to the local instance by IP geolocation. Authentication can just be forwarded to the home server via SAML, thats also where the settings can be stored and queried at login time by the local instance. SAML assertions are very scalable, and there needs to be no global login server, since every users login query can be forwarded to his home instance, where his profile is loaded. The full search index could be put into a blockchain that every local instance joins - every instance crawls their area and publishes new results to the chain. You seem to know very little about how the internet works, yet you accuse me of raging.
That the foss community can manage things like that has been proven for years. Debian mirror server network works in a similar way (they run their own loadbalancer ofc), while being cryptographically secure. And if you wanna see a federated login network like i described in action, just go to https://pubs.acs.org/action/ssostart
All these parts i described are existing technology and in global use. The combination is not, but there is nothing that would prevent a foundation from implementing search like this.