Old urban myth. Byrd was referring to an ice-free region of the mainland called the Bunger Hills. I've seen photographs of it in a nice coffee-table book on Antarctica. It is by no means representative of the vast remainder of Antarctica. The aerial reconnaissance is hard to reconstruct; maps of the routes taken are not readily available, and the research was superseded by the following International Geophysical Year. There is no "ice wall," unless you want to call it the beginning of the 2-kilometer thick cap that covers the rest of the continent.

There is no prohibition on private travel. Legal restraints can only be exercised by the government of which the intended travelers are citizens. You can book a tourist visit to The South Pole (the actual pole) by any number of agencies. For reasons of safety and international agreements relating to protected areas, permits may be required. (When presented with these facts, Antarctic mythologists usually retreat into defeatism by not even trying to find out anything.) Since gyrocompasses become ineffective near the poles, it is a very risky flying environment (not to mention having potentially dangerous weather).

Here is an interesting account of the Byrd expedition (Operation Highjump), which was both more and less than one is led to believe. https://www.smithsonianmag.com/air-space-magazine/operation-highjump-18223476/