Santa left both unwrapped and wrapped for a number of years...mostly unwrapped gifts near the empty cookie plate
I found out that Santa/St Nick was inspiring and "with" my mom during shopping for hard to find items (cabbage patch year for example) when I blurted out loud that Santa's handwritting "Love Santa" looked just like mommy's
I personally like gift bags and cheap-o tissue paper all scrunched on top. Gives the pretty picture essence of time cosuming paper wrap, but you can still fold away and reuse the bag the next year. Last year my niece was more into the paper than the gifts inside...whole you can give a baby the best gift and they want the box or your empty water bottle sitting next to it!! LOL