Governments do have free speech rights. Those rights may be limited compared to the rights of private citizens, but they do exist.
Authoritarian governments, of course, have all the rights they manage to grab. Look what Russian officials say. I suppose that North Korean officials can say anything their bosses tell them to say, any public opinion notwithstanding.
The government is protected for speech related to its duties, in clearly spelled out ways.
..which is a different claim than "governments have no free speech rights." They do, just limited in ways that individual free speech rights are not. But even individual free speech rights aren't absolute.
Language matters.
I predict a case about this will reach the Supreme Court in the next few years.