OK, this is crazy. I (think) I have finally found a clue as to what is causing the web server to slowly choke on too many open connections.
This thread for 2001 (!) explains the problem, AND IT IS STILL HAPPENING 24 years later. #FreeBSD
https://bugs.freebsd.org/bugzilla/show_bug.cgi?id=25986
I'm rebuilding the server with the patch from 2013 (!!) to see if it helps.