I have flown several types that have pitch trimming effected by a variable incidence tailplane/horizontal stabiliser but have never come across one that has the problems alluded to for the B737. This has set me thinking ....
I think that the first issue is predictably following an AP disconnect. If an aircraft is in trim, the pilot will control the aircraft in pitch intuitively. However, if it is sometimes in trim, sometimes not and by differing amounts then it will be very unpredictable and there will be a finite reaction time before the pilot starts to make any required inputs. I have flown one aircraft that was longitudinally statically stable and did not have any follow-up trim post engagement. Therefore, if a disconnect occurred and there had been a speed change since engagement then the aircraft would be out of trim. However, it was still predictable because knowledge of the speed variation from selection fitted a mental model of the aircraft's flying qualities and it was consistent.
The second consideration relating to out of trim conditions is whether the pilot is attempting to maintain a constant pitch attitude or to generate a pitch rate. If he/she is attempting to maintain an attitude and has the elevator authority to do so (i.e, requires less than full elevator), the force can be held initially and then stabiliser trim inputs made. The force will reduce and the control column will need to be moved back towards the trim position. However, whilst retrimming the pilot will subconsciously be correcting any errors in the desired pitch attitude via yoke displacements rather than forces and so the aircraft's response will still be predictable.
I believe that the problem aspect is when the pilot is trying to generate a pitch rate but full elevator deflection does not give him the required rate. In the case of a low manual go-around with an AP disconnect following an AP nose up trim input and a nose up pitching moment due to power, a nose up pitch rate may exist even if the pilot applies full forward yoke. If nose down stab trim is applied simultaneously, a nose down pitch rate should eventually be generated, and due to the resulting steep nose up attitude a nose down pitch rate is now required to regain the desired pitch attitude. This pitch rate will be generated by both the elevator and the stabiliser but the pilot has no sense of the relative contributions of the two controls. When the desired pitch attitude is achieved, aft movement of the yoke may not stop the nose down pitch rate if too much stabiliser trim has been applied and even with full aft yoke the nose may continue to pitch nose down. I suspect that this is why the Boeing FCTM advice posted by Centaurus advocates the use of bank and not simultaneous elevator inputs and stabiliser trim; the key is in the first line when it refers to "pitch rate".