Technically the internet also doesn't have "global search" but people are able to get along just fine most of the time.

This is how offline social networks work, and it might be fundamentally the only way social networks end up working. If each instance can't filter what it receives, then spam is too easy. If every message is globally flooded, the system scales as O(N^2) and is easily vulnerable to DoS.

AT solves these problems. Even if AT turns out to be a bust, they have an excellent architecture.

AT works by the use of global relays which see everything.

Sure, but it shows global replies, it provides global search, it's not O(n^2), it's not easily DOSed, and it's highly amenable to spam filtering, which are the issues you raised.

It's true that this solution doesn't work for private posts and DMs, but the n in O(n^2) is much smaller there, so I don't think it's as much of an issue for personal data servers to communicate directly in those cases.