This is exactly how effective censorship works. For example, what most people don't understand about Chinese censorship is that the foundation of their system is that everything is attributable to someone eventually. So they start by targeting anonymity. Then when something they don't like is published and gains traction, the originating party and the major distributors are punished -- sometimes very publicly. The chilling effect is that people will learn to self censor. Oh and they keep the rules really vague so you always err on the side of caution.

CBS self censoring is basically the same thing.

The Chinese government can then say "What censorship?" or "It's rare" and now the FCC can do the same.

Playing whack-a-mole is not a good strategy for censorship. The chilling effect of self censorship is the winning strategy.