That's actually an interesting idea. Always capturing, perhaps at half-fidelity to save wear and tear on the storage card and full quality when they activate for a call. I don't know why I've never seen anyone suggest this.
I can see legitimate grounds to switch them off. E.g. talking to a witness, or informant, who doesn't want anything on record.
It should be properly noted in their records however, and viewed with suspicion if overused.
FYI, good police actually love the cameras. They vastly cut down on false accusations, or at least nip them in the bud. It's also, apparently extremely satisfying and effective to show someone exactly how they came across. It really runs in a drunk tank hangover.
Then they can submit a formal request with detailed documentation to request payment for those times their camera is off. Still seems like a perfect idea.