I’ll second it. Again, if we go outside the wilderness “national parks” I would maybe throw in Gettysburg.
The most remarkable aspect of the Grand Canyon is the geology - almost 2 billion years of rocks exposed. I spend a lot of time hiking here, route finding is some of the most challenging and interesting in the world as you hike through the geology, with routes intimately related to the way the different rock layers erode. There’s really nothing like it anywhere else on earth.
Yellowstone, Yosemite, Grand Canyon, and Carlsbad Caverns
I agree with the OP’s top three and throw in Glacier as the forth.
Have you ever been there? 
It’s the most beautiful and terrifying place I’ve ever been. Just hearing your own footsteps that sound like you’re walking over a giant sinkhole scared the shit out of me, all while looking at the most beautiful scenery imaginable.
Yellowstone is pretty much THE definition of a National Park.
Really hard to list w/o describing criteria. Best for visiting, or best just because they exist?
I’ve been to Denali - not too impressed. There’s all manner of wilderness up there. Denali is a bitch to get to, and not amenable to a bunch of hiking without guides. Having said that, I’m damn glad it’s there. just in no hurry to go back.
Personally, I’d include Volcanoes. Seeing new lava ooze out at my feet is definitely on the top of a very short list of my personal ecological experiences.
Then I’d probably add Smokies and/or Everglades. Everglades, b/c they are so different. Smokies, because their popularity says something about Nat’l parks.
Re: Badlands - when we stayed there, I was disappointed that my wife and 3 kids were so underwhelmed. I thought it pretty awesome. We were staying in one of the cabins on-site. But the other 4/5 of our group had us cut our stay short one day. Now the minuteman silo west of there was pretty awesome and IIRC - is administered by the NPS. 
Not much to add but I find it ironical that Mount Rushmore has yet to be considered for the Mt. Rushmore of national parks.
Heh. 
It’s a National Memorial.
That’s because it’s a National Memorial, not a National Park (among, possibly, other reasons).
It kinda sucks.
No, I have never been there, and I am sure it’s all that when one is there, but just from photos I’ve seen, I would never be able to identify it as Yellowstone particularly. The Grand Canyon is immediately identifiable in photos even to people who have never been there.
First ‘Mt Rushmore of National Parks’ when Mt Rushmore is a National Park, now extending to ‘Teddy Roosevelt of National Parks’ when there is a Theodore Roosevelt National Park. It could get confusing at some point .
I find it hard to rank the parks we’ve been to. It depends not just what you like but how you experience it. For example I went to both Yellowstone and Arches with a pretty big party of inlaws, fun trip, but they were in a relative hurry in both places so only a few hours at Arches and a couple of days at Yellowstone, w/i 200 yards of the big SUV. The only place that blew me away on that kind of visit was Grand Canyon.
OTOH, though no big wilderness hikers by any means, my wife and I alone did some hours of hiking some miles from the car at Theodore Roosevelt (in the ‘badlands’ of western North Dakota, not as well known as some others) and Glacier this past year and those were both very rewarding, above what we got out of Yellowstone or Arches, visiting them the way we did. And in fact because of the difference in terrain I could not even 100% say Glacier is ‘better’ than TR. TR is amazing in its own way.
What, no love for Rocky Mountain NP? I’ve visited twice; the first time, I remember looking up at the towering summits and thinking, “I see now why humans used to think their gods lived on mountains.”
Teddy Roosevelt is the most underrated park in the country. Pretty country and uncrowded. It doesn’t have a ‘remarkable’ sight, but it makes you understand how the Great Plains must have been. If you’re half a mile off the loop road in the South Unit, you’re basically alone and you can pop over a hill and see a herd of buffalo or wild horses. Just a great place. Although I’ve heard it’s more crowded than it used to be. I remember the first time I went in the early 90s, we passed 2 cars on the loop. I have heard that that’s no longer the case.
I’ve actually been to Teddy Roosevelt National Park but I was just passing through so really didn’t get to see much.
Photos of Yellowstone that include Old Faithful are pretty memorable. It’s impossible to confuse it with another Park.