3 reasons that I know of - and the first is visible on your map:
Image credit NASA
Take a look at the topographical lines around Coats Land - the ascent to the Antarctic Plateau is more difficult from the Filchner-Ronne Ice Shelf than it is from the Ross Ice Shelf. It doesn't look that bad on the map, but there is roughly a 10,000ft (~3000m) elevation difference from the ice shelves to the Plateau.
The second reason was that the Weddell Sea is extremely inhospitable compared to the relative calmness of the Ross Sea. To borrow Thomas R. Henry's quotation from Wikipedia:
The Weddell Sea is, according to the testimony of all who have sailed
through its berg-filled waters, the most treacherous and dismal region
on earth. The Ross Sea is relatively peaceful, predictable, and safe.1
In fact, Ernest Shackleton's Endurance Expedition in 1914 failed largely due to the conditions in the Weddell Sea, and they never even made landfall on the continent.
Finally, there are 2 passes through the Transantarctic Mountains that make the approach much easier from the Ross Ice Shelf, as illustrated by Gordon Home's map of the routes followed by Roald Amundsen and Robert Scott. Amundsen's route is on the left, Scott's on the right.
Image cropped for clarity, original available here.
Note that Amundsen's ascent to the Antarctic Plateau is also about a quarter of the distance to the Pole than route from the Filchner-Ronne Ice Shelf. This also made logistical support (laying supply depots) much easier, as well as limiting the distance that had to be travelled at the higher, colder, and windier elevation.
1Henry, Thomas R. (1950), The White Continent: The Story of Antarctica