I'm sure they had stage bottles and they were just editing for the sake of brevity, but the way they described it was that it was important to deliver the specimen as fast as they could.
They may well have delivered the specimens to support divers at 20ft who brough the specimens up to the surface while the "deep divers" did a stop, breathing from surface supplied gas/02.
Note there were also probably safety divers in the water with them in case they or any of the film crew had issues.
But if I were the producer, I'd know that there would be a good chance that divers would watch the show, so I'd be a little more thorough explaining the nuances of the dive.
No you wouldn't.
You think that way only because you're a diver. The show was about THE CORAL not THE DIVE.
There was also a chance that boat owners would watch the show, should they have been "a little more thorough explaining the nuances" of the boat they were diving from?