I've had another look at this, and I don't think it's related to the drives at all. I tried it in the car this morning, putting the player to sleep in tuner mode, and when I woke it up again I got the mysterious behaviour - blank screen for about 2 seconds and battery icon for a further 7 or 8 but without the disk icon appearing. Since the car's engine was running, the disks weren't active, and I never used to see this, I doubt that the supply voltage really is too low (though I don't have my multimeter handy).

I wonder if it's possible that the code that checks the supply level is confused by something. I don't know how it works in either hardware or software, but is it possible that thread scheduling or interrupt masking (including servicing higher-priority interrupts) is somehow making it difficult for the player to get a correct reading for a period of time?