OK, getting farther. On a hunch I remoted into the machine and found that the resolution defaulted to 3840 x 2160, which is way beyond what the TVs can handle. So I dropped it down to 1920×1080, and now it works on the 1st display.
I can't enable the other three TVs; the PC sees them, but it won't let me extend or duplicate the desktop on any of them. When I hit Apply, Windows barks saying that it can't save the settings.
What am I missing besides a brain?