TL;DR I have an apparent fix for the problem now. See below.
The Mac was not sleeping; it was already set to never sleep, and I continued to have live SSH sessions to it, and interactions with other network services running on it. Just the NoMachine remote desktop session stopped updating.
When double-checking the energy-saver preferences to verify I had disabled sleeping, I noticed the “turn off display” was set to 10 minutes, and it doesn’t appear to be a coincidence. I already had the NoMachine server prefs set to blank the screen when a remote client connected, so whether the screen was on or off didn’t matter to me (and it isn’t obvious now whether NoMachine is actually turning the display off with the backlight powered down, or just blanking it).
But setting the system preference “turn off display” to “never” appears to correct the problem for me. I’m guessing the Mac powers down the GPU along with the screen, and the video memory stops updating, so the NoMachine server can no longer tell anything’s going on.