Comment by kelnos

I think too many of us are used to movies and TV (and Star-Trek-like scifi) that gives the incorrect view that extremely detailed information about the state of things is available.

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.