Replying to my own post to add some information. First of all, I know almost nothing about this stuff, I'd just like to rescue my ProMedia 2.1 system if possible, so please bear with me.
On further analysis, it looks like the voice coil is pretty cooked (see attachment). It took me a while to recognize this as the voice coil, I expected something bigger from looking at pics on the internet. I guess the burning/smoke was the voice coil and adhesive. Looking around, there are some opinions on the implications of a burnt voice coil, such as at sound.westhost.com/articles/speaker-failure.html#s7. According to this, discolouration in the middle or all over the coil indicates that the coil was "driven with too much audio power for too long". I almost never turned these speakers up past about 1/3 of the way, even accounting for Windows volume control. In fact, at the time of the burnout, my daughter was listening through headphones attached to that speaker, there was no sound coming out at all. Other sources blame "pure overpower", or a power surge for a fried coil. I have read that this system has fuses protecting both channels, but I have been unable to locate such a fuse on my model. Inside my subwoofer/amp I only see one fuse labeled "Fuse 1", and it looks nothing like the kind of replaceable fuse that is shown for this system in other posts on this forum (mine is square/grey-black).
Here's a nice video of a speaker/coil being driven to the point of flameout youtube.com/watch?v=KErbjC98yoM. I will note that my speaker would begin to smoke almost immediately when powered up through the subwoofer/amp, without any movement at all.
I've got a used ProMedia 2.1 satellite speaker coming in the mail, and I plan to harvest the mid-bass and put it in my satellite (which is also the control pod). I would like to hear if anyone has suggestions on how to test the other components, either in the amp or the satellite, to determine if they might be the source of the problem. As I said, a visual inspection reveals nothing, in fact everthing else is quite pristine. I want to feel safe leaving this speaker plugged in to the amp!