In a small but significant step, I decided to confirm that the Nextion communications code still operates correctly when there is NO USB connection. To do that, of course, I have to supply power to the Arduino other than through the USB (which has been its only source of power to date). So the long-term plan has been to supply it with the shared 12V supply that the motor is now using, and the (soon to come) LED 12V matrix will be using. Just a one-wire addition, the 12V from the motor board to the VCC on the Arduino. I removed all the debug diagnostics (which go through the USB to a serial terminal) and built a "stand alone" version of the code. Uploaded it (via USB of course) and then pulled out the USB cable. Still functioning - Arduino is definitely getting its power from the 12V.
The supply is slightly over 12V, which puts it just outside the recommended voltage for the uC (but only by 0.6V or so). The uC can supposedly handle up to 20V or so. Anyway, all appears to be working - the first completely stand-alone version of my televisor.
It's been playing Doctor Who in the background for the past 10 minutes or so, seems quite happy. I checked the menu functionality (which was always a bit dodgy probably because of the serial double-usage) and it seemed quite solid. Another small step - no longer needs a laptop to be connected to actually work
Edit: Has now been running for over an hour non-stop (Doctor Who on repeat) and still happy. That's the longest, by far, that the televisor has run continuously. I did a touch-test on heat; motor is warm but not hot; say 30C - the TIP122 is cooler, about 20C and the IRL540 is basically cool. The Arduino is putting out a bit of heat; I'd guess 40C. A touch warm, but not super-hot; I can keep my finger on it comfortably. So, all seems good!
Edit: I later measured the temperatures, and you can see the results here!