It's really the other way around: film crews can legally travel with the police.
This is very similar or identical to what you saw on YouTube. From http://blogs.findlaw.com/celebrity_justice/2014/04/cops-and-suspects-rights-whatcha-gonna-do-when-they-record-you.html (dated 4/18/2014)
"Cops" will soon begin a 10-week filming stretch in San Jose,
California, for the first time in the show's 26-year history, reports
the San Jose Mercury News. The "Cops" crew will hit the streets
alongside San Jose's finest to provide a window into what these men
and women do in America's 10th largest city.
More from that link:
When the officers on "Cops" arrest most of their suspects, the
circumstances leading up to the arrest are out in public. However, in
order to avoid unlawfully appropriating the images of those caught on
"Cops'" cameras, the crew asks the arrestees to sign a release form.
The "news crew" are producers who have permission from the police to follow along. As above, they ask each suspect to sign a release after the arrest. These arrests take place in public, on public streets and right-aways, where filming of the public is legal. If the arrestee does not sign the release, their face is blurred in the resulting footage that is made public.
This does not mean the producers or videographers are automatically allowed into a private residence. They need a release to go onto private property (unlike the police involved in an investigation) and as such, the resident can actually forbid the cameramen from entering the house even while the police have entered.
The footage from the body cams of the police are a different story; they are public employees of the local government and are bound by the laws of the locality. A search of Google News shows that in San Fransisco, the issues of the use of body cams by police and public access to that video is an ongoing issue; some laws have been finalized, others are bound to change.
Around the US, some localities at this point in time allow public access to the footage from police cams; others only after department or local governmental review; still others only on in as need basis for prosecutions.
...the cop is the only one in this situation who can command the
filmers to stop.
Even if the "news crew" is not a news crew and just someone from the general public, it is generally legal for the public to film the police, as long as they are not causing issues with the police and stay out of the way (re: all of the recent news regarding police shootings and the footage available on YouTube and in news sources). There is case law pertaining to that, but IANAL, so I'll let someone else outline that.
But https://photographyisnotacrime.com/ is a good resource. In the US, you pretty much have no expectation of privacy while in public.