The Republican Party
Historically the Republican party has been a big tent which deliberately sets out to include a wide range of opinions on the political right. As a result there are, roughly speaking, two main wings in the Republican Party:
Libertarians, who oppose big government , support tax cuts and gay rights, want to see an end to the drug war, cut back the military, rein in the police, and open the country to immigration. These are the people who say "just let business do it's thing". (They also detest Donald Trump).
Culture Warriors who want more military spending, oppose any controls on the police, and want lots of government action against anyone they define as "Bad", including drug dealers, abortion clinics, immigrants, and at the more extreme end, teachers of evolution, gay people, Black Lives Matter demonstrators, Muslims, blacks, Jews, environmentalists, name it. These people don't care about the Libertarian position on anything; they just want the government to go after the Bad People.
(I'm simplifying. The situation on the ground is obviously more complicated, with some people shading between the two groups, and others who just nod when one of these talking points is on Fox News without really thinking it through).
Historically these two groups have been able to get along inside the Republican Party. The Libertarians are in the minority so they don't get everything they want, but by staying in the tent they can have a bigger influence than if they were outside.
However Donald Trump's campaign aimed squarely at the Culture Warrior wing and ignored the libertarians. Many of them decided he was a Great Man, leaving the libertarians are even more sidelined than usual. The libertarians will carry on voting Republican regardless, because to their mind its better than voting Democrat. Hence Republican politicians play to the Culture Warriors because they are the people who will vote for the winners in the Primaries.
As long as social media was not an issue in the Culture Wars Republican politicians could just ignore it, or take the Libertarian line if need be. But once social media became an issue it was the Culture Warrior take on it that drove the response, not the libertarian one.
What made social media an issue in the Culture Wars were the moderation rules. Computer geeks often lean libertarian, and the people who came out on top of the Internet boom are no exception. They envisaged social media as being enlightened places where people of good will could discuss the important issues of the day and arrive at consensus. They were disappointed. Practical experience repeatedly demonstrated that this doesn't work, so the social media companies were forced to institute moderation policies and delete things they felt that their users were not going to like (because if people don't like what they read on your site they are going to go somewhere else, advertisers won't pay you, and bang goes your Unicorn).
(Aside: I'm going to stick with "moderation" here. "Censorship" means government action to suppress speech. Its the difference between "You can't say that" and "You can't say that here".)
The trouble with moderation (and censorship for that matter) is that, no matter how well intentioned you are, you always wind up looking stupid and petty (Can we call that "Johnson's Law of Censorship"?). This situation is worsened for the social media companies because the sheer volume of material, combined with a desire to keep the costs under control, means that much of the moderation is done by low-paid individuals working under extreme time pressure, and there is no effective recourse if you feel aggrieved by their decisions.
The social media companies decided that hatred and incitements to violence were off-limits, and hence inevitably wound up making decisions on them that could be considered stupid and petty, especially by the people on the receiving end. These tended to be more on the Republican side because the extreme end of the Republican party tends to like the idea of violence against the Bad People. From there it was a very short hop to "Nobody could be that stupid and petty by accident, so it must be a secret company policy to block us".
This belief was reinforced by discussion of "shadow banning". Republicans came to believe that their views were deliberately being given reduced prominence in Twitter. The trouble is that, absent a leaked memo or whistleblower, these allegations are very difficult to prove or disprove. However they rapidly became a staple of mainstream Culture Warrior thought. (The Left also complains of unequal treatment, but the Republican Culture Warriors ignore this).
Which brings us to the current situation. The big social media companies effectively control the boundaries of much of the discourse in our society. Culture Warriors don't like this because they don't like the place they believe those boundaries to be. Libertarians also regard this as problematic for more principled reasons. And so the Republican party wants to see government control of the social media companies.
Parler was another attempt at this (and a more principled one at that; if you don't like the rules build your own place with the rules you prefer). However they rapidly learned that you will have moderation, and you will wind up looking stupid and petty.