If you want to get technical about it, the Catholic Church was deeply embedded in most European governments when the Index Librorum Prohibitorum was issued, and was the official state religion for multiple kingdoms.

The Hays Code was issued by the MPAA as a voluntary set of guidelines governing the then vertically-integrated studios and theaters. Importantly, they were not prohibitions on movie content. Many film productions did not adhere to the Hays Code and many independent theaters similarly ignored it. Indeed, the body established to enforce the code did not even how the power to order studious to cut material... The only instance of the Hays Code having an impact on films was during the short Breen era, which lasted only 6 years, and this only affected films shown in vertically-integrated theaters owned by the studios. Independent theaters continued to ignore the Hays Code, as they always did.

And the Broadcast Standards was a decency code intended to mollify advertisers, i.e., other private parties, who generally did not want to be associated with certain types of content. As with the Hays Code, it was a series of guidelines, not a prohibition, and late-night shows generally flaunted the Broadcast Standards with impunity.

In other words, not censorship...

Just private acts of self-interest to protect their market position.