Would not the problem for the pilots been similar to 38 pilots losing an engine on take off? That was my first thought seeing the film. 38 pilots were trained to reduce throttle to avoid flopping over on thier back from the torque of the operating engine. Couldn't it be these guys just ran out of time?

Every multi engine aircraft has a VMC speed, i.e, the speed at which you no longer to able to control the aircraft flight on one engine! When doing through multi engine training, the one thing that is stressed so much and trained so much for, is operating on one engine! There is a "best rate of climb" on one engine in all twin engine aircraft!
If in the event of power failure, as you are cleaning up the bad engine, feathering, turning off fuel and etc, you want to be getting to that single engine climb speed. Now for reasons, sometimes beyond your control, late raising the gear would be good example, you are below that VMC speed, you are not going to have directional control over the aircraft and to keep it right side up, you would have to reduce power on good engine in order to maintain directional control. That may have been what happened, but don't think so! If that had enough time to climb to 1,000 AGL, they certainly had time to raise the landing gear after takeoff, which again leads me to believe they shut down the only good engine they had!
When viewing the recorders, engine #1 was the "failed" engine and shortly there after, #2 stopped producing thrust, so my guess right now is they just plain shut down the wrong engine when going through the emergency procedures!
Be aware for non pilots reading this post, VMC speed and stalling speeds are two different speeds, the aircraft stalling speed being the less of the two.
The whole point of my original post was, if this aircraft had a larger rudder, It would be able to maintain directional control down to a much lower speed!