. . . Conservative Christians, as a political force, only came into any significant power in the 1980s and 90s, and by that point they were already very much out of favor by any of the media elites, and so had very limited impact, and on those areas they did, it's because they were part of a large coalition involved in a moral outrage.
If you're referring to things like the Hays' Code and such prior to the 1960s, I'm sorry, but despite how you like to push anything that the modern left dislikes into the "conservative camp", that was very much not the case when you actually look at the historical movements.
Firstly, the American Conservative movement, of which Conservative Christians are a part of, did not truly become a thing until the 1960s, the very time in which the Hays' code was collapsing. Even if we go with the precursor movements to American Conservative Christians, the Fundamentalist descended Churches, those had no political or social clout in the early to mid 20th century, since they were still rebuilding from getting summarily ejected from the academy. No, the bulk of the push for controlling the media originated from general society as a whole, but mainly focused in the Mainstream Christian Churches (which, I again must note, had been recently captured by the decidedly more progressive Modernist Christians). In point of fact, early 20th Century Progressive Christians were a decidedly censorious lot who sought to improve society from many top down activities. The Temperance movement, for instance, was a big part of Progressive movement, and it was Progressive president Woodrow Wilson who saw the literal apex of Speech Control in US history, and he was decidedly not someone whom fit into the mold of modern Conservative Christians.