They absolutely were/are private, even if they used a publicly owned medium to distribute their content. Other than an hour of public-service airtime on Sunday mornings, and lax FCC enforcement of a decency standard, those private TV networks can do/say anything they want.
You wouldn't put quotes around that word for cell-phone companies using licensed bandwidth, or airlines using public airspace, would you?
The broadcast companies are subject to FCC requirements that they must broadcast government emergency notifications. They do NOT have the power to shut down access the way twitter just did.
And they are not private. The public is legally entitled to receive anything broadcast over the radio spectrum. And as I and another poster have already pointed out, there are government licensing and carriage requirements involved with TV and radio broadcasters.
You wouldn't put quotes around that word for cell-phone companies using licensed bandwidth, or airlines using public airspace, would you?