I imagine getting status updates with intermediate search results, and I annotate each one with 'warm' or 'cold' and maybe add some more search terms into the hopper to forcibly narrow or expand the search.

Central search indexes like Google are not going away. There are client-side metasearch interfaces that combine Google results with other sources. Those other sources can be much slower, including human responses. You would still have your synchronous sub-second response from centralized search, but there would be asynchronous results from decentralized search.

This exists today, e.g. when you post a question on HN or a messaging app, asking other humans for answers not available in public indexes. Most of the world's knowledge is not public, it's obscure and may only be of interest to specific niche audiences.