Half as Interest's Sam Denby explains in his characteristic mad way why commercial aircraft often fly over the North Pole, but never over the South Pole. The main problem is how far the plane can be from the nearest airport at any given time. Modern aircraft have improved the range problem, not the problem of flying over Antarctica.
"Basically, every commercial twin-engine plane is given a certain rating for how far it can be, at any given time, from a suitable diversion airport. This is to ensure that there’s always a runway that a flight can divert to in the event of an emergency… This rating is called ETOPS, and it’s measured in flight time. …before 1985, all twin-engine planes were ETOPS-60, meaning they could only ever fly within a 60-minute radius of an airport. As planes got better, ETOPS numbers went up."
That said, the closest airport to Antarctica is quite far away, well outside the ETOPS (Extended Range Twin Engine Performance Standards) ratings of most commercial airlines.