Thank you very much!
I did an intensive debugging session, and use the WP File manager plugin as well, since you had it pre-installed. I found an actual issue within the query results comparision script. It was a tiny mistake, but caused big trouble. I could not replicate it on our test servers as it was specific to configuration and to a result size above a limit.
I made a tiny change to the search algorithm to correct that mistkake. If I’m correct, the results should be ordered correctly by their relevance, and the title matches should be atop of all of them.