I think it depends on your families tradition of when you open gifts. we open them in the morning since the kids are all excited and stuff. But when the Santa gifts take over the morning and it's a hassle to get the kids away from them to open the other stuff then I say maybe wait and bring them in differently.
Last year and the year before Santa left his gifts in one of the kids bedrooms and that kid slept with a sibling or grandparent. I was up all night wrapping anyway. I closed the door and left them unwrapped. It was a Kangaroo Climber and a bike so it would be awkward to wrap. Last year it was a bike and tricycle.
This year Santa brought a trampoline and we tied huge jingle bells on the spring joints and big floaty ribbons all over too. The kids thought some gifts someone placed on our porch last night without labels telling who they were from were the gifts from Santa so they didn't even miss getting a gift. I even pointed out that Santa ate the cookies and drank some of the milk, still no "What did Santa leave for us". I finally had to go outside to do anything to draw their attention to the yard and they saw it and didn't figure it out until I said "Look what Santa brought to you guys". The temp was in the low 30s with a vicious wind chill so they didn't last long on it.