Yes, I agree that enforcement of moral norms requires setting arbitrary limits on behavior that aren't necessarily objectively defensible. Every law and ban regarding the dissemination of goods and services involves a slightly illogical compromise between a partially elected moral authority and the citizenry.

There isn't really a clear answer to your questions about whether or not "we" would continue to censor video games, popular media, and TV shows if limits were introduced about what minors can see on the internet. It would depend on the compromise reached between the citizenry and the enforcement agency that tasked with enforcing the bans. In some regards, I feel that American life is far too censored and supervised, but in others I feel that it is far too liberated and unrestricted. I happen to believe that there should be more content limitations on entertainment and social media, and less content limitations on political speech. If I were ever to meet with a large group of people who shared my beliefs, we would probably take political action to enforce our will for society.

I suppose that doesn't really prove or disprove your assertion that censorship laws would inevitably lead to more extreme censorship, but I think it's important to recognize that there are already many and forced censorship laws in the US, and altering them isn't really the craziest thing.