AA Cali - A striking example of how important it is to keep all your skills brushed up - especially those to do with the complexities (on occasions) of the automatics. Unless you understand how to use them, and use them properly, they will rise up and bite you.
They are tools, to be used to an end. If they don't perform the task required, dump them. If they lighten the load at a time when there is much to do and a short time in which to do it, then use them. Don't scorn them because of some outdated idea that only wusses fly on A/P, and "in the old days we used to hand-fly 200 tons of aircraft through hurricanes with total blackout on the ground and land on a 60 ft runway in less than zero/zero visibility - THAT was flying, son!"
I'm not quite sure why Eastern came up - that was nothing to do with hand-flying.
In the case of the BM 737, the first thing the skipper did was to disconnect the autopilot. He therefore occupied himself unnecessarily with handflying instead of analysing the problem and solving it. He was far less able, therefore, to carry out a full scan of the engine instruments. Who knows - he might actually have spotted the vibration indicator going off the clock...