Welp, a third time replacing yet another bad capacitor in the TV. Now awaiting kids' report to see if it actually fixed the problem again. The TV powered back on and is showing a picture after my repair so that's better than nothing.
For anyone wondering how one actually knows to replace a bad cap, the TV would predictably power-cycle and reboot itself whenever a video went to an all-white screen.
Open it up, look closely, and you'll see a tented cap.