We always played Post Office. I still have fond memories of the game.
It has no real rules, and I think it was designed by my parents mostly for chaos control with a horde of excited children.
Each of us in turn got to be PostMaster, while everyone waited in a circle at their assigned “addresses” (like, at the desk, sitting on the sofa, etc.).
The PostMaster would select a gift from under the tree, read the tag silently to him/herself, and then “deliver” it with fanfare and name announced to the recipient.
That recipient then became PostMaster to deliver the next gift. No one could open presents until all the gifts had been delivered.
Oddly enough, we had lots of fun disguising who the gift was for, and holding everyone in suspense as we each wandered the room pretending to locate the right address and recipient.
It was actually more fun than the presents were.