I don't think a bow that is cut so the arrow is centershot is inherently more accurate than one that is not. However, more care needs to be taken in tuning arrows for the non-centershot bow. An arrow correctly tuned for a non-centershot bow will bend around the bow and fly straight, just as though the bow were centershot.
If a bow handle is strong enough, I can see no reason not to cut it so the arrow is centershot, other than some competition rules, as was noted. However for many self bows, the handle is not strong enough. In The Traditional Bowyer's Bible, John Strunk contemplates whether it is worth the risk to cut a 1/4" platform for the arrow rather than providing an arrow rest on the side of the bow handle, and concludes that "in my estimation, it is not worth the risk."