Recovering a "dead" lithium battery

[jdurand]

It seems like a good idea for the battery management circuitry to just shut down and stop drawing power before the pack gets to the point where it can't be recharged with the normal method, then wake up when plugged in. I'm not an EE so may just be showing my ignorance.

While true, that is not normally the case. Almost all battery packs use one of only a couple of different chips or the microcontroller in their product that also does other things. Essentially nobody has any incentive to do it differently.