Well, have you ever seen a movie or read a book w/ Santa in it?
I'm thinking... and the only movie (or book) I can think of where Santa gives wrapped gifts is in one of "The Santa Claus" movies w/ Tim Allen. In the movie, Santa gave wrapped gifts to adults at an office party.
I have a plate commissioned by Coca-Cola displayed on our mantle. On the plate is the Coca-Cola Santa Claus, & he is in front of a Christmas tree w/ unwrapped toys - he's playing w/ a helicopter that he brought to the home.
I enjoy reading everyone's traditions! And, like you, I'd never heard of wrapping Santa gifts until the DIS.