Okay, sports fans, I've restored 4.3.9 on my private server. Yay!
The problem was that I had updated ruby. That caused local extensions to not be found. The solution was to do this:
gem pristine --all
and enjoy a cup of coffee. The errors stopped, everything was happy again, & I restarted the servers.
I'll pin this one on both servers for a while to spread the solution for "Source locally installed gems is ignoring [whatever] because it is missing extensions."