It's helpful to look at that case in the context of the time, which was pre-New Deal, more federalist, and the Bill of Rights applied narrowly to the Congress. It was about a state (Ohio) having a censorship board, not federal censorship.
The argument wasn't even made that it was a violation of the first amendment (which would have only applied to laws by Congress, not states). The argument was more about things like whether it was a violation of interstate commerce to have to have different versions of a movie for different states. They did argue that it violated the Ohio state constitutional right to free speech.
(reading that again I discovered https://en.wikipedia.org/wiki/Mutual_Film_Corp._v._Industria... ; the idea that movies were not counted as free speech for several decades in the US may come as a surprise to other HN readers)