I THINK I detected that the autopilot works a servo TAB (trim tab if you will).
It doesn't. Then, the rest of the reasoning is flawed.
There is a good reason why NO airplane with hydraulically powered controls have trim tabs, for example: They don't work!!!
Hydraulically powered control surface don't free-float until finding the equilibrium position where the hinge moment is zero, like in you Cessna or my Tomahawk.
They are "rigid" systems that do not respond to external forces: If there is no flow of hydraulic fluid (which is controlled by hydraulic valves), they will not move no matter the external force (unless the external force was such to overcome the hydraulic system, which is never the case otherwise the hydraulic system would not be able to move the control surface against these loads, which is exactly what they are designed and intended to do).
(Therefore, on go-around you really need two tabs giving you your nose-up inputs, otherwise it happens a bit too slow / insufficient.)
(The final stages of a normal, average ILS involves SMALL control inputs...a single autopilot and single tab is more than sufficient, whereas a go-around is a significant pull up)
PP's argument was exactly the opposite: That one single AP channel lacks enough initial* pitch DOWN elevator authority to counteract the pitch up moment created by the underslung engines when they go full coals. (*initial because the elevator trim, or movable stabilizer in the 737, which is also controlled by the AP, will quickly take over, but it will take a couple of seconds for that trim wheel to spin the required number of turns).
Like any true trim TAB, if you adjust the TAB and nothing is holding the controls, the yoke will move ("back drive" = the $20 way to say that during your 172 walk around- you move the elevator up and down, the yoke goes in and out (as opposed the in-flight opposite where the yoke moves the elevator, the elevator moves the yoke).
That doesn't work in the 737 as long as at least one of the hydraulic systems is pressurized, for the reason explained in the beginning: you will not be able to move the elevator with your hands because you have an hydraulic actuator that is full of uncompressible fluid and with its valves closed.
The "back drive" that Flyboy mentioned, which according to him is the same backdrive that PP mentioned, would be something that artificially moves the control columns when the AP moves the elevator, which I presume would be something related with the artificial feel system, yet another thing that is required because the hydraulically powered elevators don't, can't back drive any force (i.e can't provide force and position feedback on the control columns).