Ironically, the security cameras on people's private property that cops are choosing to put tape over before possibly committing crimes are closer to "full recordings" than what body cameras offer.
Body cameras are sold with features that cops and their bosses like, like features that exist for the purpose of keeping the public from accusing cops of violating their rights or worse.
That means that many body cameras are designed to only record short 30 to 60 second segments of video after a cop feels that "something bad" might happen to them. Many of the cameras also come with convenient on and off switches, too, and some come with features that allow cops to delete videos.