This kind of stuff should not be left unanswered:
an aircraft that was overweight
It was about one ton overweight. A lot on a C172, 0.5 % on a Concorde. It had no influence on anything. Not good, but not a factor.
, had maintenance issues,
Are you talking about the missing spacer? Or the intermittent fault in one out of three flight control channels, which was an acceptable defect in the MEL? Again, not good, but no factors.
out of trim limits,
Huh? Where did you get that idea from?
All available data says it was inside limits.
taking off downwind,
Wind was light and variable at the time. The tower reported 8 kt tailwind may or may not have been present on the runway during the takeoff (it would have been present at the wind sensor, which is not the same thing), but analysing the aircrafts actual performance during the takeoff indicates it was not present.
with a non-captain commanded shut down of an engine
The captain ordered fire drill at the same time when the flight engineer announced "shut down engine 2". So the captain did order the shut down.
Hardly. Engine 2 never recovered from the surges, and was producing no more than 15% of thrust when it was shut down. Shutting down engine two put the other three engines in contingency, increasing the thrust on these engines. They probably got more thrust when they shut down engine 2, not less.
Could engine 2 have recovered and started producing thrust had it not shut down? Maybe; I doubt it, but maybe. But with a fire alarm from engine 2 it would have been shut down in short order anyway. Keep in mind that engine 1 was less affected than engine 2, it was not shut down, and it still surged a short time into the flight. Engine 2 would not have ran for long even if it wasn't shut down.