1. AF1 is a 747. If you've never been on a 747 (or A380), then you might not appreciate how smooth landings are in general. It's a lot of airplane with a lot of momentum (not to mention a lot of electronic assistance for the pilots).
2. You say "The notion that the Airforce One pilots are somehow special with special landing skills strikes me as fanciful." I don't consider that notion fanciful at all! In fact, I'd expect they have special landing skills. The AF1 pilots are not just some random commercial or charter pilots, they are Air Force officers, hand-picked from the best of the best. In addition, not only do they have to take-off and land the president's airplane, they also must be able to take evasive action IN A FREAKIN' 747 in the event of a hostile encounter. So "special landing skills"...I'd buy it.
EDIT: Also, reading the original Quora answer, the particular landing mentioned was on the Shuttle Landing Facility, one of the longest runways in the world! I'd wager the pilot could've practically let AF1 roll out the landing and had room to spare.
But are you advocating that pax in an A380 approaching big airports with nice weather just walk around as they please? And why not? Because the risk during takeoff and landing is elevated, compared to cruise flight, isn't it?
It's a tricky question, granted, but I'd still say, "principiis obsta", nip things in the bud, don't let safety standards slide, buckle up.
(Under part 91.107 [1], where part 91 are general rules for general aviation, and that rule does not hold for operations under part 121 (air carriers/airliners), 125 (planes with more than 20 pax), 135 (commuter and on-demand flights), but they all re-introduce it separately [2][3][4], so yeah.)
Either they're all criminals, or the rules on military flights are different.
At any rate, I'd think it's rather prudent to sit down and put a seatbelt on - the rule's not just for fun, but because accidents can occur. The notion that the Airforce One pilots are somehow special with special landing skills strikes me as fanciful.
[1] https://www.law.cornell.edu/cfr/text/14/91.107
[2] https://www.law.cornell.edu/cfr/text/14/121.311
[3] https://www.law.cornell.edu/cfr/text/14/125.211
[4] https://www.law.cornell.edu/cfr/text/14/135.128