The Auto throttle does different things dependent on the mode it is in and the phase of flight.
As a general rule, though, I would suggest that in Airline flying, your "Point the jet use thrust to control speed" is what is normally used on approach.
As an example, the most usual approach in an Airliner would be an ILS- if done with the automatics, the AP would use pitch to follow the Glide Path, Roll to follow the Localiser, and the A/T would be in SPD mode, adjusting thrust to maintain the selected speed.
There are modes when this is not true, however- for instance if wishing to make a flight-idle descent, Flight Level Change mode simply reduces the thrust to idle, then uses pitch to maintain selected speed.
A pertinent factor about THAT mode is that after a few seconds at idle, the A/T mode changes to "Hold", which means it is basically dis-engaged and in standby. This allows manual changes to thrust to be made if you wish to vary your ROD.
In the Asiana case, it SEEMS that what the crew did was manually use pitch and roll to follow the approach path, EXPECTING the A/T to manage the speed, but the A/T was in fact in "Hold", and thus not doing so.
Hope that helps somewhat!!