I grew up with a Santa and the gifts were always wrapped including what was in the stocking.
My husband is Mennonite and never had the joy of Santa until he married me :-) . I was his first Santa and his presents were wrapped in special paper. I got a lot of harrassment over this from an in-law for pagan idols until I explained the "goodness of Santa and the joy of giving" to them and told them to shut up. Can you tell I might not be the most popular in the crowd?
My children's gifts from Santa have always been wrapped in a separate paper that was only used for those gifts. I don't think my kids will admit to ever finding anything early. They didn't want me to know they figured out Santa at all. I knew they had worked it out when they started filling MY stocking.