NavList:

My understanding is that it's mostly about diversion airports for two-engine aircraft, which cover most low and moderate volume international routes. If your aircraft has an engine-out, you should land soon. That's the rule. The Arctic and the Antarctic are both empty, but the difference is that the Arctic is surrounded by land which includes some big airports with long runways (Cold War legacy, in part).

I actually have a former student who lives in Perth, Australia. I haven't seen her since she was in eighth grade (I'm only seven years older than her, so... do the math), but we are "Facebook friends". She announced a few weeks ago that she and her husband are taking a trip to Buenos Aires. That's the perfect trans-antarctic great circle case, but they don't fly that way. It seems that the diversion rule is the main factor. 

I was inspired by your question to see if I could find any more details. The Wikipedia article on polar routes has some good details. They make the interesting point that weather-routing rules out the great circle route even for a case that looks good on paper: "They will not fly over the South Pole, but around Antarctica taking advantage of the strong winds which circle that continent in an easterly direction. Hence, the "westbound" flight from Buenos Aires would actually travel south-east south of Cape Town, over the southern Indian Ocean and on to Perth, while the true "eastbound" flight would also head south-east south of Tasmania and New Zealand, over the South Pacific and on to South America." Winds beat geometry.

Frank Reed