After several more attempts at holding down the power button, each followed a mysterious lag of several minutes of not touching the device only to have it restart on its own but still not allowing me to log in, I tried a new tact. After the holding down the button for a long time then waiting for the device to restart suddenly as if acting on its own accord, rather than swiping to the sign on screen, I held down the power button again. This last time, that action was followed by the appearance of the familiar "Power Off" option. I swiped it, and the device shut down. When I restarted it, all was right with the world.
I have no idea what initiated this weird behavior, but I hope this doesn't portend something more ominous in the near future.