Rest materials function like a crude plunger button. They don’t affect tuning much, if at all, but they do affect where the arrow hits, both right/left and up/down. Using a softer rest material functions the same as reducing the pressure on the plunger button.
I typically use a Martin rug rest on both strike plate and arrow rest, because I seem to need to move my POI to the right, and like you, would like to reduce my point on. It moves my POI about 3-4” to the right at 45 yards, and I assume it drops my point on about the same amount, although I’ve never measured it. Not much of a change, but I’m glad for whatever help I can get.