Growing up on Christmas morning someone was chosen to be Santa. No we did not dress up in a Santa suit and go ho-ho-ho. What we would do is pick the presents out from under the tree and hand them to the person whose name was on the present. I got to do it a few times and I enjoyed it. I would look at the presents under the tree and carefully pick a present out. I would be kind of random at first and not care so much about who but eventually I would try to pick presents out for everyone so no one felt left out.
I think the point of having someone direct the present handing out was so that it lasted. Instead of letting 5 kids rip through the presents for their own and just open them willy nilly. By directing the present opening we all got to see what everyone got and make the magic of opening presents last a little longer.