The real reason is that the people running those AIs that they put so much money into to get where they currently are are well aware that people can just run their own models locally and cut them out of the picture entirely so they are attempting to monopolize the hardware needed to run it under the guise of getting ready to expand into the future.

Though the storage I think is more about wanting to keep a log of each interaction with their model for training and other ways to profiting from it (openly or covertly).

Like the whole "ai output isn't copyrightable" benefits them because they can just straight up use whatever output their models generate and it will be perfectly legal.