You seem to live in a world filled with unreliable sources. But if they are actually reporting their route and the distance traveled which is quite easy to do in this day and age of gps, then the distance they traveled would make the ice ring impossible, and also would prove that the continent is smaller than the average diameter of the route they traveled. So actually viewing the coastline they circled would not be necessary to set an upper limit on how big Antarctica could be. Also, any ocean race like this has checkpoints along the route that every yacht must pass in the correct sequence in order to prevent the kind of shortcuts and cheating you suggest might have occurred.
As for a more reputable source, there is the one that involved teams of scientists from all over the world, also with several stops and ports along the way that they visited in sequence, including stops on the actual continent: http://spi-ace-expedition.ch/
It appears that it took 3 months for those people to make that journey. How do we know that they didn't go around the Antarctic Rim?
The winner made it in 74 days. The greatest distance covered in any 24 hour period was 537 nautical miles which was a new record for this race. The winner averaged more like 330 nautical miles per day ( 13.75 knots) over the entire race. And all of this includes the travel time and distance from France to the latitudes at which they circumnavigated and back up to France.
So sure if they cheated and turned on their motors (which they did not have as that would slow down a competitive ocean racing yacht) then maybe they could have traveled more like the 70-75,000 nautical miles down to and around the ice rim on the unipolar map in 74 days. However, the yachts in this race could not ever make a pit stop as that was against the rules, so there is no way they could have carried enough fuel to make it that far. A large motor boat could easily burn 2 gallons or more of fuel for every nautical mile at the necessary speeds to cover that much distance....so say a minimum of 150,000 gallons of fuel weighing 900,000 pounds (the yachts in this race weighed more like a total of 15-20,000 pounds). Yeah, that would not really work. These yachts relied on the wind and there is no way they could average such high speeds 24 hours a day for 3 months. Their top speed is around 30-35 knots, so even with extreme wind conditions 24 hours a day for months on end, they still could not cover the distance.
Either the entire race and all of the participants in the eight races that have been run are all part of this big, dumb and pointless conspiracy to fool us all into thinking the world is round, or else they simply sailed around Antarctica. Here is the winner's yacht under sail:
These yachts are amazing feats of engineering and are going so fast already that it is pretty much torture to ride one for weeks on end:
http://www.yachtingworld.com/extraordinary-boats/close-look-hugo-boss-alex-thomsons-vendee-globe-2016-92713