It is not just the 777. The 787 and 747-8 have the exact same logic for A/T wakeup.

The reason is the auto throttle can't wakeup to provide speed protection in FL CH SPD, VNAV SPD, TO/GA because the pitch is controlling the speed.

You can't have the speed controlled by both pitch and auto throttle.

They had V/S selected, set missed approach altitude, and at this point were very high. PF selected FL CH to get down, but the A/C started to climb to the MA altitude.

PF pulled thrust levers to idle, and A/T mode went to hold.

Initially in the descent he was more or less following the F/D.

Once the PF got low and started to shallow out he was no longer respecting the F/D.

As the FCOM states, AFDS speed protection is provided through the elevators in FL CH SPD, VNAV SPD, and TO/GA. But you have to follow the F/D.

The PM had the F/D off.

If the PF wanted to ignore the F/D guidance, he should have turned off the F/D.

If there is no F/D active, than the A/T wake feature is now available.

The design is good, and in fact really the only plausible way.

How any training department can claim that the A/T will wake up in all situations is irresponsible at best.