Felix,
You have probably accurately quoted the documentary (I missed it), however their explanation is quite illogical.
All the A/T ever does is move the thrust levers. Fuel flow to the engine is controlled by the Electronic Engine Controller (EEC). The EEC receives several inputs, a prime one being thrust lever position; note that A/T on or off has NO effect on EEC commands.
If fuel flow reduced when the A/T was disconnected, this could have been due to a manual movement of the thrust lever. Or, it could have been coincidental timing, with the EEC finally realising at this stage that fuel supply was excessive, or the engine RPM finally winding down enough (due damage) to reduce vibration, or....
My point is that disconnecting the A/T cannot in itself cause a decrease in scheduled fuel flow to an engine. Which brings us back to the question ... why did the vibration stop (about the time the A/T was disconnected)?