In a more consumer friendly architecture, the bank would send the consumer a list of the current organizations that have a recurring charge 'relationship' and ask the consumer which ones should be transferred to the new card. It takes it from automatic, to semi-automatic, but much more consumer friendly. Needs lots of security around it, of course.
Cancelling a Netflix account is a lot easier compared to something like a gym membership. Are people really cancelling their entire credit card instead of going to a web site and clicking a few buttons?
That applies to all recurring credit card charges.