My impression has been that my kids wouldn't just be "just as happy with half as many", they're actually
happier with fewer presents.
If I watch the body language of my kids during these present-opening extravaganzas, I can see so many cues: The annoyance at being pushed to keep opening presents before they're done engaging with the one they just opened. The saturation point where they're nearing some stage of psychological numbness after umpteen presents. The stress of being pressed to show some semblance of gratitude toward all the people who are giving them presents even after they've reached that point. The increasing piles of toys giving increasing opportunities for squabbling with the other kids in the room over toys, or feelings of envy and jealousy around all these material possessions, etc.
And then, after that's over and everyone's gone home, they end up enjoying the stuff they have less over the long term as well, because having such a large collection of toys creates such a high distraction environment. It's hard to realize it when you're in the thick of it, probably doubly so when you're a kid, but owning lots of stuff is mentally exhausting.
So this year, a few weeks after Christmas, we talked through all that with our kids, and we agreed to start curating the toy collection. They helped us pick a selection of stuff to put in the closet, and, whenever they want to, we'll rotate some stuff. It turns out that, a couple months later, they're still declining to do any rotation: They want to stick with the stuff they already have. They're busy finding increasingly creative ways to play with what they've got, and I honestly think they're getting a greater feeling of satisfaction that comes with the accomplishment of designing new games with the things they already have, and of finding cool new things to build with the one construction toy they chose to keep out.
It really seems to me like, by taking their toys away, we gave them back their fun.