Censorship is part of existing as part of a society though. There are people who seek to actively hurt others using words and media, and the whole idea of a state is to (pretend?) to act in the common good. At this point the internet gives everyone a platform (look at us right now, there's a near 0% chance we would have ever spoken in physical real life) and makes a TON more noise that makes finding the signal harder, which means both sides can find plenty of examples of the other behaving badly to "whatabout" whenever someone points out their own bad behavior.
TL;dr- Everyone wants censorship, even if they pretend they don't, so let's not pretend it's actually either side. If anything, your point just proves that Dems/leftists might have more power in the US to influence these things, but that doesn't prove they have more desire to influence them than the right.