"Depends on the situation at the time of engine failure"..correct,
During every return to lease flight we do this
CRUISE AT OR ABOVE 37,000 FEETA. DUCT PRESSURE
(1) Check that duct pressure regulates between 34 to 50 psig _____
B. ENGINE ACCELERATION & HIGH STAGE BLEED CROSSOVER
(1) Engine IgnitionCONT
(2) AutothrottleOFF
(3) Maintain stable flight at cruise altitude.
a) FMC N1 page select CON ____
(4) Engine #1
a) Set No. 2 thrust lever to MCT.
b) Rapidly retard No. 1 thrust lever to idle.
c) Check for crossover indicated by sudden duct pressure increase _____
d) Allow RPM and EGT to stabilize.
e) (Timed event) In approx one second advance the No. 1 thrust lever toward MCT. Set thrust lever approx 1 knob width below engine No. 2 to avoid overboost.
Start timing when thrust lever is advanced and stop timing at 90% N1.
f) Record acceleration timesec.
g) Engine should accelerate smoothly and without compressor stalls ____
The reason I post this is to simply emphasise that the aircraft does not fall from the sky or lurch hopelessly one way or the other and a small control column deflection and rudder input is sufficient, the big difference being we are mentally prepared for the event.
If you loose an engine in the climb or turn and do not revert to basic airmanship and problem diagnosis there is a likelihood of the incorrect actions being taken. The A/T should not retard the TL in the climb unless the target altitude had been captured in the MCP +/_ the limits of VNAV or LVL Change, VS does not count here as climb in VS the AT is controlled by the MCP speed. During an engine failure in the climb yaw would be fairly significant, but again, there would be no commands to the AT servos unless some signals had been given to the system either in terms of MCP speed reduction in VS or "Capture" in another mode.
An A/T servo failure during TL retard should be spotted by the crew but is not a reason to have a jet upset and certainly not regarded as so "time critical" that diagnosis cannot be performed. To be honest, the turn and slip is more of a comfort thing, the yoke displacement is a more natural indication and does not lag as much as the ball.
Apart from EFATO, at a safe height, the excessive use of rudder trim is not recommended as unless you are terrain critical and need to continue the climb, as normally you would simply stop the climb and after the diagnosis trim of the QRH recommends it.
The reasons for this accident are yet to be established and the whole engine, TL, AT debate is speculation, it could just as easily be a jammed or broken flight control.