"I often wonder if municipalities could copyright their broadcasts like"
Generally? All of that IS copyrighted. Under the Berne Convention, which the US signed onto around 1976, all copyrights are automatically assigned to the creator from the moment of creation. However.
The US is fairly unique among nations in that all government copyrights have always been considered to be property of "We the People", so a government agency itself cannot copyright anything they create, and does not own the copyrights to anything they create. Those rights belong to the American people. Although, we tend to allow aliens to reproduce the materials as well.
Some major agencies have had to have that explained to them by the USAG because they just couldn't understand who the boss is.
And that pesky first amendment? Means anyone can shoot their mouth off, but they still can't yell "FIRE" in a theatre just for fun. And they can't cross other lines, like disclosing national security matters. Police action IN PROGRESS would count as one of those matters. You can shout about it--but you'll still be charged with "interfering with a police activity" and that's the end of that.
No, this is unfortunately simply legal blackmail, they know that Broadcastify can't afford to spend what it would cost to bring a defense in court. And they know they have no jurisdiction over Broadcastify, or else they'd have stated what it was. Their only proper course of action is to encrypt what they don't want made public. TFB if that makes eavesdropping over radios and phones impossible, that's their right, under all the laws from the FCC on down.
Legal blackmail, OTOH, needs to be smacked down. It has become a real problem in the US over the past 20-30 years, getting worse every year. Someplace with bigger bucks (like the Electronic Freedom Foundation or the ACLU) might be willing to take up a countersuit and make some good out of this.