Human explorers aren't necessarily terribly rational. A bunch of ordinary human pride is involved.

The base on the actual South Pole, far on the interior of the continent, isn't McMurdo where much of that documentary was filmed, it is Amundsen-Scott, and it's named for two teams of explorers who first reached that pole in the same summer, Amundsen (whose team reached it first and returned alive) and Scott (whose team was second and all died)

Scott's plan was crap. To a considerable extent that view is hindsight, but even at the time Scott must have known Amundsen's plan was better. Certainly by the time they reached the Pole and found that they'd been beaten to it, he will have been sure. By that time he was in extreme danger, slowly starving and with a long trek back to any permanent shelter - it would have taken excellent luck (which he didn't get) to make it home alive, and in any case he'd been beaten to the pole.