Flying with two engines and taking off without an engine in a loaded aircraft are two very different things. A lot more thrust is needed during takeoff than after.
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 speed (past a certain speed, you only abort if the aircraft is literally unflyable - for everything else, you take the aircraft & emergency into the air and figure it out there). 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.
There's also a very big difference between "Engine failure: something has damaged or jammed enough components that the turbines are no longer spinning fast enough to produce thrust or drive the generators" and "Engine failure: the engine is no longer attached to the aircraft, which is why it is no longer producing thrust". However, both things are reported in the cockpit as ENG FAIL.
(Un)fortunately, cockpit warnings prioritise the what (so to speak) and not the how or why. On one hand, this makes decision-making a lot simpler for the crew, but on the other...well, in rare cases the lack of insight can exacerbate a disaster. Depending on when exactly the engine gave out, this poor crew might have been doomed either way, but they might have been able to minimise collateral damage if they knew just how badly crippled the aircraft was. And there was a very similar accident to this one (actually involving the predecessor of the MD-11, the DC-10), American Airlines 191 - one of the engines detached from the aircraft, damaging the leading edge of its wing in the process, causing that wing to stall when the crew slowed down below the stall speed of the damaged wing in a bid to climb. If they could have somehow known about the damage, the accident might have been avoided entirely as the crew might have known to keep their speed up.
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.
In this case however, with the wing already on fire (the engine is below the wing, so flames coming out of it would be visible behind and under the wing, not in front), I'm afraid that even if they had managed to take off, the fuel tank would have exploded or burned through the wing before they would have had a chance to land. Actually, this looks similar to the 2000 Concorde crash...
In fact, for awhile (maybe still the case), the #1 killer of skydivers was single engine failure on takeoff from the jump plane (and similar aircraft failures), not accidents ‘while skydiving’.
