The autopilot handles engine out (multiple engine out actually) extremely well. Autopilot is available in alternate law I (but not II or direct law). You have to mentally separate the autopilot and the auto thrust. They are two separate systems, and many failure modes will affect one but not the other.

The autopilot, in either ALT mode, or APP mode will be using pitch to control altitude (ALT), or to control the glideslope (APP). Speed control via pitch control is not used in these modes...speed is controlled by power only, by the A/THR or by manual thrust lever positioning.

I'm not exactly sure which of the many 'trips' was causing the autopilot to disengage, but seeing it disengage as you hit a well marked speed bug would instantly alert you to the fact that it didn't like that speed. The problem in this instance is that the normal response of just going a bit faster wasn't desirable (you want as little energy as possible at touchdown). The approach speed chosen would still have had a large margin over the stall (20% or so), so flying manually and accepting a couple of knots below the target would be perfectly reasonable.

Whilst overweight landings are normally flown using the autopilot to touchdown, the autopilot cannot handle automatic landings in unusual slat/flap configurations (lots of reasons), so the landing would have ultimately had to be done manually. Automatic landings also tend to use up a couple of hundred metres of extra runway...another reason not to do it. When manually landing, there is no fixed rule as to when we disconnect the autopilot..anywhere below 1000' would be quite normal.