You can't use an airplane to simulate one-sixth gravity.

Actually I think this is an anti-conspiracy argument. It's well established that you can use an airplane to simulate zero gravity by nosing over in parabolic flight path. The motion picture Apollo 13 demonstrated that such a process could be used for accurately simulating the absence of gravity for purposes of fiction.

It can also be used to simulate one-sixth gravity by flying in a slightly shallower climb. The Apollo astronauts trained this way [Reports11b, 76].

The "vomit comet" was used to stage the zero-gravity segments of the Apollo missions, just as it was for the motion picture Apollo 13.

The video from the command module during the coast phases of the voyage was broadcast seamlessly, with no breaks or "cuts" from one camera to another. That's because there was, in fact, only one camera. The parabolic flight path method only works for about 30 seconds at a time. It would have been quite impossible to use this method to produce a seamless video presentation lasting several minutes. But it works fine for producing a Hollywood feature film were a "take" only lasts a few seconds anyway. And this footage could be intercut with scenes filmed on the ground using more conventional stagecraft methods for simulating weightlessness.