I have worked at some companies that require the AP to be on for an RNAV SID and now I work for one that does not. In my experience, the AP can do just as bad of a job as a hand flying pilot. This is largely due to pilots not paying attention to their increasing ground speed and/or shallow bank angles.
If there is a sharp turn after departure, the FMS calculates when to start the smart turn so the flight director will roll the airplane out exactly on course. If the FMS does that calculation assuming your ground speed is static (let's say 160 kts shortly after takeoff), but during the turn your ground speed goes up to 250 kts, you will roll out off course with or without the autopilot engaged. Similarly, if you are hand flying, and your bank angle is too shallow, the same thing will happen.
When I use the autopilot, I make sure to keep my ground speed constant prior to and during the turn by keeping a high pitch angle. If I am hand flying. I do the same, but also make sure I start out with a 30* bank right away and shallow it out as required.