In case this helps anybody who observes the same warning: After digging around in the code (NB: I feel like the practice of catching various exceptions and turning them into “end-user-readable” messages is quite harmful for debug logging; oh how I wish I had full tracebacks or possibly even stackprinter-package output) I got the suspicion that putting my search terms in quotation marks might help. It did not help in the basic search but it did help if I used advanced search for text-fields like document content.
So now I got search results again. But to be honest, I am not sure if the quotation marks were the solution to my problem and whether the above mentioned warning was actually the root cause. Because even with the quotation marks, the search results were incomplete. So my suspicion is that I was the victim of the search indexing breaking before it is complete, like many others have reported in this forum (Search reindex doesn't work - #5 by joerg.sichermann, Search not working anymore and others).
Long story short, I went back to Django/DB as search backend (described here: Search Backends with Small Servers - #3 by DocCyblade) for now and have a working search again.