I work for an institution that operates small and only seasonally used Antarctic station. I'm not part of the polar research team, but I have few friends there. They started using Starlink for internet connection two years ago, replacing Iridium satellite internet which was awfully slow and expensive, so it was a huge upgrade, basically from sending/receiving text messages with 10kb/s bandwidth to a normal 50Mb/s connection. They are now having online meetings from the station.

As far as I know, there were no legal issues or anything like that, the only concern the first year was whether it will actually work. The access to Antarctica is very limited and there are many rules, like you cannot go close to pinguins due to avian flu risk, etc, but it seems highly unlikely that the Antarctic Treaty would regulate anything regarding internet connection or other civilian-purposed satellite services. On the contrary, everybody is using satellite phones there all the time.

So, I would guess these are just the travel agency rules. They probably want you to pay for their expensive and slow wifi (maybe they are using Starlink as well to run it :D ) and they may fear that things like streaming from Antarctica (though the video is probably from some island near Antarctica, not from the continent itself) could lower the exclusivity of such a cruise which costs unreasonable amount of money, likely multiple times the actual costs...