Funny, because I thought just the opposite - I didn't realize so many people wrap gifts from Santa. Honestly, I think my parents started the unwrapped because my brothers and I always woke up Christmas morning by 5am and they didn't want to get up then so we were allowed to play with our gifts quietly until a decent hour. Wrapped stuff was always necessities like underwear, socks, pjs, pants, etc. We rarely got excited about the wrapped stuff.
Is the picture of your son, your first? In general I found my kids were completely overwhelmed with unwrapping (birthday or Christmas) and ended up just opening 2 or 3 and then wanting the boxes rather than the gift and then we had to unwrap the rest later anyway so it was an easy decision for us to continue the Santa straight-from-the-workshop toys.