Diogo,
The distance that a bow is or is not cut to or past center should not effect your accuracy as long as the arrows are properly tuned to the bow. It has been my experience that a bow cut slightly past center is more forgiving of spine issue and there for can accept a wider spine variance and be a little easier to tune.
As far as putting something under the rest and sideplate to create a pivot point, it can improve your accuracy. What this does, is create the smallest possible area that the arrow makes contact with the bow. What this does is to minimize the effect of any bow torque you may be inducing during the shot execution. I do create such a pivot point on my bows, by placing a small piece of toothpick or strip of leather under the shelf and side plate. I line this pivot point up with the deepest part of the grip. If we all had absolute perfect form then this would not be needed, but this of course is very difficult to obtain and maintain over all types of shooting situations.
You are never wrong for continuing to look for ways to improve your shooting or form. To me it sounds like you have either a form issue or arrow tuning issue to look at as the cause of accuracy (by way of your description I would lean toward tuning). Your arrow is already heading off the mark when you are releasing, either caused by tuning or form, it is just not far enough off the mark to noticed until the long shot distances. My question would be what type of arrow tuning method do you use and at what distances do you tune at. An arrow that is not properly tuned and missing its mark by a few inches at 10 yards, could be missing your mark by many inches at 20 or 30 yards. A pivot point may help to improve this, but I would look harder to find the root problem.
Chris