I'd argue that trying to solve problems caused by an untrustworthy entity by trusting them to operate a system that determines whether to trust them or not is inherently a bad a idea.
The punishment for turning off the camera will always be less than if the camera had captured some type of obvious wrongdoing. These cameras will be like the locks on your front door. They're there to keep the already honest people in check.
The key is accoutability. At least this way we can see who is breaking rules: and certainly, there'll be less of a punishment for turning off a camera: but at least there'll be something. That will necessarily dissuade some people; maybe it won't dissuade everyone, nothing does, but all that will remain is the kind of person who desperately wants to hurt someone that they'll happily take some punishment for it. That's a minority, and even they will face some charges. It's a start: and the crucial thing is, it's a shift in thinking as well.
There's no firm divide between honest and dishonest. Certainly, some people are honest, some people are dishonest, but I'd say the majority are in the middle ground: dishonest when there are no repurcussions, no danger, and honest when they might have to face consequences.
When your country has average case lengths such as New York City's estimated 900 days for a criminal case to be heard in court, then you end up keeping an awful lot of data. In addition, the statute of limitations on many crimes is at least a few years, sometimes decades. To delete evidence that could be pertinent to cold cases seems like a bad thing to do.
Not every case goes to trial: sometimes confessions are given, for example. Further, the primary use of body cams is to hold police accountable: they'd still be successful if footage was only kept more than, say, a week/month if the police officer is suspected/charged/accused of brutality. Plus, not all of the footage will be needed: no one cares about a officer singing to themselves while driving around. If you want to be thorough, have someone take minutes of the non-major film, and only store the interactions: when the police officer finds or pursues or arrests someone.
Still, with the advances made in cloud computing, the existence of flash drives, even your extreme case is far from unwieldy.
Theory aside, body cameras are in use in the UK, and save £400,000 ($611,000) a year. The benefits seem to far outweigh any hypothetical negatives.