Earlier quoted context omitted.
Taking off with one engine inoperative (on a multi-engine aircraft, obviously - you aren't going to get anywhere with your only engine gone) is completely normal/within design parameters, albeit undesirable. In fact, it being normal almost certainly contributed to the scale of this accident, since a single engine failure during the takeoff roll isn't considered enough of an emergency to reject the takeoff at high spe…
> The crew wouldn't have had any way to know that one of their engines had not simply failed, but was straight-up gone with their wing on fire to boot. I don't know about the MD-11 itself, but other aircraft from that time period have sensors to detect and report overheat and fire in various parts of the aircraft, including engines and wings.
The notification in the cockpit is likely nothing more than "ENG 2 FIRE" or similar. That could mean anything from "the fire is minor enough and we're at high enough speed that it's significantly safer to take off and then make an emergency landing", to "the engine has exploded and the wing is on fire and catastrophically damaged, so even though aborting takeoff now is dangerous and will likely cause us to overrun the runway, trying to continue would be worse".
It's a judgment call by the pilot to guess which of these is the case (or any possibility in between), and given the probabilities of various failure modes, I think it's fair for a pilot to assume it's something closer to the former than the latter.