I'm sorry you, you must have misread DataCorp's filing but the argument is that a limited contractual agreement to broadcast a channel (e.g. programming selected and owned by someone else) does not give DISH unlimited copyright over the shows used in that programming.

It's akin to a radio station claiming copyright infringement for songs that they are only licensed to play. The copyholder isn't the radio station.

DataCorp doesn't assert that copyright is absent but that DISH is not the copyholder. For my reading that seems accurate unless DataCorp submitted false details to the court (something I doubt).

Or is DISH bonkers, using the wrong tool for the job?

It sounds like DISH is arguing based on the channel, not the content.