Sounds right, for continuous operation. If it were only teh drill that wasnt' working, I'd suspect just a surge current causing the BMS to shutdown.
If you can keep a voltmeter attached to the BMS output wires while you've got it in the drill, and then watch the meter while activating the drill: If the BMS output goes to zero then it's shutting off to protect the pack from something--overcurrent or cell-LVC. If it doesnt', but the drill doesnt' do anything, there's something the drill needs that it isn't getting from the pack.
Regarding the charger, if you're using the original charger, it probably won't operate without the thermistor, because that is a really big safety thing with the NiCd stuff--without it, the charger doesn't have a way to know the NiCds are' getting hot enough to catch fire and stop pushing current thru them. (it's simpler to do just a thermal check than a delta-V check, which makes the chargers cheaper to make)
Regarding the drill, maybe it also has actual electronics in it that are powered by the pack, and read the thermistor for overheating in use, in which case you'd also need to leave it in place.
If you've removed it it's easy to add back in, it's not polarity sensitive. Just wire it between ground and the unused pin on the pack's terminal block.
If it's not that, then I don't know--maybe the pack's original wiring was backwards, so the negative was on red, and positive on black, and there are diodes or other polarized components in the drill.
The old NiXX stuff I have here is much simpler than that, and doesn't care whcih way I hook up a LiPo pack to run it.