It is not "freedom of the press" when newspapers and others are allowed to say and write whatever they want even if it is completely false!
What makes Alvarez surprising is that the Roberts court had generally rejected free speech claims when the institutional interests of the government were at stake, showing deference when the restrictions on speech were for the military or in schools or in prisons. In Alvarez, the court struck down the Stolen Valor Act, notwithstanding the government’s claim that it was important for the military.
Second, Alvarez is one of the court’s most emphatic statements that false speech is generally protected by the First Amendment and it is for the marketplace of ideas, and not for the government, to decide what is true and what is false. The protection of false speech, of course, is not absolute. There still can be liability for defamation and punishment for false advertising. But the six justices in the majority were clear that speech cannot be punished just because it is false.
Put most simply, Alvarez stands for the proposition that there really is a First Amendment right to lie.