I understand what you are saying, but it isn't your speech the government is restricting. It's your recording of the officer's speech.
You don't get it yet. The right to record is linked to your first amendment PRESS rights - your right to gather info before disseminating it.
See, the right to a free press refers to a TECHNOLOGY, not a job description. So your use of "press technologies" is fully protected even if you're not a pro journalist - hence you have a right to own and use the computer you post here with, and a camera with which to gather info you disseminate on forums like this.
There are police agencies such as the NYPD who are still claiming a right to decide who "the press" is. Such positions are also in a state of collapse and the switch is fully "on" in the courts to reading the 1st Amendment "freedom of the press" as a technology as opposed to a job description.
For a more detailed look: