Interesting thoughts about the caching of the single-word searches. I hadn't imagined that they would do that, but it seems to make sense, as the search function in the Apple Music desktop app does do a "live search" as you're typing, which indicates that JavaScript is likely at play (would this be AJAX?). As you mentioned, the Web app and desktop app may be passing common or popular searches on to an Apple Music Web server that handles such, but less common searches are perhaps passed on to a different service. The query to the latter service is apparently failing on Sonoma. This suggests that they changed the server-side code from Sequoia, but failed to update the client-side query in Sonoma to match the new server-side code.
Maybe.
I also confirmed today that it's not just multiple keywords searches, but also uncommon single keyword searches that throw the generic "Something went wrong. Please try again later." error.
Also, about Feedback Assistant, I used that for several cases, but it seems that Apple never updates the status on anything there, even though internally they may be referring to user feedback. They probably just designed the tool thinking it was a good idea (or based on complaints from users around the world and a request from upper management to get their act together), but the devs likely don't have the time to update the feedback themselves. I surmise this based on my own experience in the software development sector. With larger firms and projects, the devs themselves don't usually take on the role of client-facing dialogue and responding personally to feedback—they just don't have the time. Apple doesn't seem to have linked their own support organization with the Feedback Assistant tool, which is why customers like us see the disconnect and experience the defunctness of it all.
Anyway, it's nothing we can do anything about, except to complain using the Product Feedback and Support channels, and assume that Apple will eventually post a fix in an upcoming update for Sonoma... unless we want to go all the way and upgrade to Sequoia, of course.
