Why wouldn't they provide footage? They already let us know everything that happens in there, so it won't hurt them any more. It would allow the American people to be more involved in the court proceedings and could pull more news time to things that actually matter. Besides that, many people would be interested in watching big cases.
The day of the procceeding, they release the written transcripts. These are accessible to anyone. In addition, they publish an audio of the proceedings the friday after. This is transparent enough, annd we do not have to risk judges playing for the cameras. The supreme court proceedings are already transparent