McMurdo has starlink. South Pole doesn't, but not due to technical reasons from starlink's side. From what I understand when they tested at Pole they noticed interference with some of the science experiments, its possible they will engineer around that at some point but for now starlink is a low priority compared to ensuring the science goes on.
I forget the exact distance, but its something like 5 miles from pole that they ask groups traversing to turn off their starlink.