The privacy issue is that an agent of the public is intruding into private spaces. The body cam footage is just revealing in a more obvious and exploitable way that the problem already existed. Whatever the body cam sees is already being recorded by the mark I cop eye.
I imagine that the privacy issues would be less severe for the recordings where cops knock, present a warrant, and then enter to execute it, because that is the due process instituted for the public interest breaching into private spaces.
Agents of the public should be doing their duties in public.
As for the automating, I'd use glyphs easily recognized by machine vision to mark different areas in the cop shop. Stencil-paint a glyph on the walls of the bathrooms and locker rooms, and the camera can automatically tag that video as not immediately available for public release, and blur an area surrounding the glyph for automated redaction.
If people are concerned about body cam video of themselves being released automatically without redactions, they can put up glyph stickers in their homes, or wear glyphs on their clothing. A public-private toggle switch on the cam might help, but is susceptible to bad user input or insufficient usage training.
Without some kind of explicit marker, AI may never be able to determine the transition points between public and private spaces, and certainly could not deal with adversarial attacks against that classifier. At worst, over-glyphing or inappropriate wearer inputs could exhaust the video-review resources.
But I think that it is in the public's interest for all calls to the emergency lines to be public, regardless of the potential privacy issues. Instead of having the 911 call center employees worried about covering their asses in case they make a mistake, support them with good training, checklists for the most common issues, a second tier of higher-trained crisis managers to join in for non-routine calls, and maybe an on-demand pool of expert support, such as the psychiatrists, lawyers, or explosives disposal technicians.
Leaving the postmortem analysis of an emergency only to officially authorized people allows cover-ups or deceptive re-framing to happen.