Indeed, the posted excerpt is correct for the NG, however if you fly an automatic approach the rudder servo works parallel during approach, rollout and go-around until another roll mode is selected.
Are you saying, during automatic approach, trim works in parallel... or yaw damper... or both?
I was unable to find any link between rudder trim (or YD) and the A/P in my training notes.
Thanks.
Cheers.
NSEU