Not a stretch, considering this is their intent. See in particular the 2nd point from 17 U.S. Code § 101 - Definitions
To perform or display a work “publicly” means—
(1) to perform or display it at a place open to the public or at any place where a substantial number of persons outside of a normal circle of a family and its social acquaintances is gathered; or
(2) to transmit or otherwise communicate a performance or display of the work to a place specified by clause (1) or to the public, by means of any device or process, whether the members of the public capable of receiving the performance or display receive it in the same place or in separate places and at the same time or at different times.
The purpose is to prevent revenue loss by the recording company who could host their own public event (or selling rights thereof). It would be an impossible argument to make that the police (or anyone) playing a song on their phone is reducing the ability of the record company from gaining revenues from hosting their own local event.
Well, that is the difference between the rule of law (there are tribunals for deciding whether the police are breaking the law or not) and the rule of incorporated companies: you are screwed if they decide it...
The author of the first public drinking law, in 1979, said on the law's purpose:
>"We do not recklessly expect the police to give a summons to a Con Ed worker having a beer with his lunch". [0]
Yet today, if you have a beer in the park at lunch you will get ticketed, or at least a warning. The lesson? Do not rely on discretion and restraint to make up for an overly expansive letter-of-the-law.
They are pointing out that the codified law is applicable, not what you claim was the original purpose.
(I don't agree with you that the original purpose was restricted to protecting revenue. The original purpose of copyright law is to codify the moral rights associated with creative works. Moral rights are fundamentally a property right connecting the work to the person. When we hear that an artist tries to prevent their works being distorted by being associated with a certain political campaign, this is an appeal to this kind of right, not revenue.)
It's not really about the purpose - it's about the rights. And it's rights in the plural - not just right to revenue - but also the exclusive right to public performance.
