At 6 miles out, only 10 feet of the target would be hidden by curvature.  They claim that 24 feet were hidden.  So why?

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The last observation, where they say "24 feet," isn't a level line measurement. You can't lump in that 24 feet with your 10 feet calculation. It's apples and oranges.
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They threw out the element of what is "level" and went to whether they could see the chopper using a telescope and a camera. The telescope and camera are shown to have a data display of some kind but what that is and what it is used for are not explained. Very non-scientific and appeals to ignorant viewers by way of theatrics only.
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We have no idea what the elevation of the laser beam would have been when the chopper did its yo-yo action by the "pyramid." (They never explained or showed any "pyramid.") The chopper pilot probably told them he refused to fly to a place where he would have to risk looking into a laser beam to position his aircraft. But they wouldn't let that inconvenient detail get in the way of THEIR DOCU-DRAMA.
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That was part 1 of a 2-part series, "Episode 2" (q.v.) being 54 minutes long. Viewer comments for the first video are in the thousands, some of which ask repeatedly whether the robot voice belongs to Stephen Hawking, then others accuse this Hawking voice of deception because he's dead. The question is resolved entirely in the 2nd video (below) because they show Hawking sitting still in his wheelchair in a movie studio staring blankly into space where they position the camera to appear as though he's looking at the camera. Creepy. He couldn't move his eyes. But he somehow managed to feed his insatiable porn appetite!
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So now you know that it's being presented as a quasi-entertainment project that Hawking envisioned, where they get 3 volunteers to run around playing footsie opening mysterious crates like Christmas presents in remote places ostensibly shipped by Hawking's whim to deliver laser, telescope, camera, scale models of earth, moon and sun, using helicopter, truck, boat, walkie-talkies, generator/air compressor, lake, Bonneville salt flats, road flare, a Star Wars style drawing stick that leaves light trails in the sand, more theatrics and the narrator voice of Stephen Hawking provided by Microsoft.
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The man uses his Star Wars drawing stick to draw a graph in the sand by way of which he somehow has used TWO DATA POINTS (the 500-foot and 3-mile boat sightings) to develop an overall CURVE that by implication represents the contour of the earth. This is not left to your imagination because they bring up an earth curvature diagram out of nowhere that says this very thing.
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They manage to spend almost as much time on the Foucault Pendulum as they do on the laser-lake project. IOW seeming to attack flat-earth by vague implication, but not openly, directly and specifically. This is argumentation by implication.
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It ends with Hawking taking the opportunity to promote his new book (which ended up being posthumously) where Hawking hawks his own personal version of heliocentrism, going to the stars, galaxies and superclusters of deep space. It's only a vague implication that billions and billions of stars (shades of Carl Sagan!) some of which have planets attached, makes earth comprehensible as if it were one of the sand grains lying on a beach somewhere.
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So it's not about crushing flat-earth at all. They don't even mention flat-earthism. It's all about selling Hawking's book. You're given the opportunity to order a DVD of his book! (Too bad they couldn't say "signed by the author" because he couldn't use his hands. Plus, he wasn't alive anymore.)
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It's a thinly-disguised infomercial.