In a multi-autopilot a/c, with an engine failure, why not fly the approach with the A/P.
But in the 737 Classic, the autopilot-flown single engine instrument approach, is neither fish nor fowl. The remaining throttle is operated manually and so are the rudders; while the ailerons and elevators and stab trim are on autopilot. In the simulator for example, we see the situation where the pilot changes the thrust manually to correct a speed trend. In turn, he then has to adjust the rudder position to correct for yaw.
Unless the correct amount of corrective rudder is fed in to counter yaw, the control wheel then turns automatically to hold the ILS localiser. This causes inevitable flight spoiler operation and thus unwanted drag. The whole event can then turn into what is beautifully described as a horses's arse. And that is because of crossed controls.
Even on a single engine autopilot flown circuit, it is not uncommon to see the aircraft turning (say) on to base leg with the PF using the heading select function of the autopilot. Meaning the control wheel turns but if yaw is allowed to occur due to incorrect rudder input it becomes quite interesting to see the control wheel turning in an opposing direction. Aircraft turning left under yaw and wheel turning right trying to correct the yaw due incorrect rudder pressure.
On the other hand a single engine circuit or single engine ILS flown manually throughout, presents no problem to a current and competent pilot because there is no mixing of manual input and autopilot input.
Personal opinion only, of course