After a couple of test runs tonight, I can now see that I was wrong in my first statement when creating this thread. The PID function does indeed actually "center" the rudder, at least a little bit. According to the graphs, the PID output decreases when error decreases, even if there still is an error. Good! 
Thank you all for your input!
What I have been thinking about is that I might give the autopilot a lot less authority over the rudder than what was initially my idea. Maybe just +/- 20 degrees of rudder angle or so. That should make it easier for the PID to narrow in on the desired course, without overshooting all over the place.
Another thing, very visible in Test run 2 & 3, is that the PID output value starts off way out of centre. Its starts at 16,62 and keeps that value until there is a error to correct. The problem is that if it is starting off with a value like that (full rudder port) there is definitely going to be an error to correct, and the boat is going to take a proper detour before its on the right course again. I need to wind up the PID controller output to about 90 degrees before the PID Output is somewhat centered and can be mechanically connected to the rudder tiller. Is there anyway of making the Output start off in the middle of its extremes. Or I might need a wrapper around it to wind it up.