For both DH & me, growing up, gifts from Santa were NOT wrapped. We both also had presents under the tree from our parents that WERE wrapped.
So, that's how we've continued things w/ our own children.
Our children receive gifts from Santa which are not wrapped. Instead, everything from Santa is set up in 3 "displays" = one for each child. If Santa is bringing a toy, the toy is unboxed & put together. Santa brings the "big" gift plus 1 or 2 smaller gifts plus some other things (a book, movie, winter scarf, etc.) to round out the display.
Items in the stocking are not wrapped either, &, most of the time, there's always something like a stuffed animal or similar sticking out of the stocking.
Santa comes after everyone goes to bed on Christmas Eve, &, on Christmas morning, everyone rushes downstairs to see what he's left.
There are presents under the tree from family. I keep a spreadsheet & decide what's going to be from Santa & what's going to be from DH & me. Once I'm done shopping, I wrap the presents that are going to be from DH & me & go ahead & put them under the tree - hopefully, well before Christmas Eve!
A little off-topic - Each child has the same number of gifts from Santa & gets the same number of presents to open under the tree. And I try very, very hard to spend the same amount on each child - or, if I'm not spending the exact same dollar amount, the gifts have the same "value" amount, if that makes sense. Even though they're different ages & one is a girl while the other 2 are boys, I try to get them similar things or things that match up.
As they've gotten older, this has gotten harder & harder. This year, their lists are impossible. DD hasn't asked for anything that expensive, plus she hasn't asked for the same amount of items for which her brothers have asked. For instance, DD has asked for decorative pillows for her bed, while our animal-loving younger DS has asked for several different stuffed animals - that's easy to match. But there's nothing on older DS's list that "matches" pillows or stuffed animals. Older DS has asked for a particular model car kit, & younger DS has asked for a remote controlled jeep which, again, are easy to match. But there's nothing on DD's list w/ which I can match. I've got blanks all over my spreadsheet, & it's driving me crazy! LOL!