I can see not wrapping a few things, but I think not wrapping takes some of the joy out of Christmas morning.
If you don't know any difference, then there is no joy or letdown of not having them wrapped. Santa never wrapped when I was growing up but he did at DH's home. I thought that was strange but wanted to incorporate some of his traditions since he was livng in my state and we spend a lot of time with my family. I must say, I love Santa wrapping gifts. It makes the morning last so much longer. My 3 kids are very good at taking turns and opening their gifts one at a time. They love watching each other open the gifts to see what they got. Santa does make sure to use paper that has Santa on it and a few years ago, he started used computer generated to/from (i.e. return address) labels which helps the elves make sure each kid gets the same number of gifts!
When the kids were small, I had a ritual of waiting until Christmas Eve to wrap everything after they went to bed. Made it hard when I discovered I needed batteries or ran out of tape, though. (one year I finished up using band-aids!
) Now my kids are adults and usually just want gift cards, so I don't get to wrap as many gifts. But I make the most of it with the grandchildren!
I just love watching them tear into the gifts. For me, the anticipation is almost as fun as the gift!

and doesn't want to create the trash and waste. I still wrap everything though, I'm sure she probably thinks we are all evil!
. Mom and dad gifts are wrapped.
