Absolutely not. Body cams mute the first part of the audio for this exact reason. Privacy is important.
The muting you observe of police footage isn't of the first part of the audio, it's the prior 30 seconds from before the record button is pressed. They have a constant buffer going, as things can happen... unexpectedly.
This caught a cop in Baltimore; he wasn't aware of or had forgotten the feature. The 30 second buffer caught him planting drugs, then faking the finding. https://www.npr.org/sections/thetwo-way/2017/07/20/538279258...
Side note: It took years to charge him (https://www.baltimoresun.com/2020/03/09/caught-fabricating-e...) and he served no jail time for trying to send an innocent person to jail (https://www.wbaltv.com/article/officer-testifies-in-own-defe...).
I just want to clarify that it only buffers the video. The way you worded it still doesn't explain why the previous 30 seconds of audio isn't included in the buffering.
When the button is pressed is when audio recording is started and the previous 30 seconds of video buffer is prepended to the live recording.