Why wouldn't the pilots know the engine was on fire immediately & abort?
Once a plane surpasses V1 (decision speed to safely stop within the remaining runway) the pilots decision is made. It’s likely the engine fire happened after they reached V1 and had no other choice but to take off.
The plane could have flown with just its right engine and vertical stabilizer engine(this plane type has 3 engines), but it seems as though right at takeoff, the vertical stabilizer engine suffered a compression stall. That reduced thrust in that engine and left the plane without enough thrust to gain altitude.
Plane eventually rolled to the left which is the side that had the separated engine. Either the left wing stalled first probably caused by damage to the flaps from the engine Fire which would have reduced the lift in the left wing causing it to stall first at that speed or asymmetrical thrust from only having one functional engine.