There is no automatic extinguisher function, and automatic fuel flow cutoff only occurs in a few specific situations (most involve engine starts). There are actually some good reasons for this. An engine fire warning at or after you’re committed to takeoff is handled in the air, usually after reaching 1000’ above ground level, because the focus needs to be on flying the airplane without distraction, and because often times that engine is still producing some usable thrust for a short time. The idea is that the steps to contain a fire and isolate the engine should occur when the pilot directs, after the airplane is safely under control and away from the ground. An engine fire/separation causing a structure fire is so exceedingly rare that the odds are better just maintaining control of the airplane for the first few minutes. A transport category airplane with a failed engine is a real handful to fly. Even a brief lapse of attention can spell disaster.
There shouldn’t be any fuel pressure drop to the remaining engines. During takeoff on a 3 or 4 engine airplane, the fuel tanks are normally isolated, with each engine drawing from a different tank. Each tank has its own set of fuel pumps and each engine has its own engine driven fuel pump. Also, the fuel lines have one way check valves to prevent reverse flow.
You’re absolutely correct: nothing in the circumstances softens the horror. I can’t even imagine the suffering of the affected families.
Most aviation accidents result from a chain of events - anything from construction, maintenance, crew, etc. It will take a frustratingly long time for the NTSB to find all the answers.
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