And even assuming the footage isn't automatically public, what about the evidentiary value against the house's occupants? Currently, police need a warrant to execute a search of someone's residence, but there's a "plain view" exception: if while the officer is in the house anyway (either because the occupant let them in, or because they entered due to exigent circumstances like a violent crime in progress), if they happen to see evidence of a crime in plain view while there -- a gun, drugs, whatever -- they don't need a warrant to use that as evidence. If the officer has a camera on the whole time, that potentially means the officer could use not just what they happened to see in the moment, but retroactively whatever can be seen in the footage after the fact, which likely significantly expands the extent of plain-view evidence that can be used without a warrant. Some have asserted that video-recording itself constitutes a fourth-amendment search if the officer is someplace where the person has an expectation of privacy, though I don't think this question has been adjudicated.
Also, in two-party-consent states, capturing body camera audio might also require either a warrant or consent for the capturing of at least some kinds of interactions.
I think body cams are net-good, but it's more complicated than it seems at first blush.