Is there anything conservatives love more than pretending they are being free-speech martyrs and being depicted with their mouths all duct-taped? It's a fetish. But no, Josh, "America" is not being "muzzled", anymore than, say, when Trump is impeached, all of his voters are being impeached. It's called personal responsibility. Conservatives could look into it, if they wanted to.
First of all, Josh Hawley represents Josh Hawley, not "America". The little stunt of trying to throw out my vote here in Pennsylvania was pure ambition, even if he wants to pretend that all of his Missouri constituents are so, so concerned about the constitution of our commonwealth. Josh Hawley needs to shut up, but that's not "cancel culture". That's shunning, because he fucked up. Dozens of police officers were injured and people died because Josh Hawley is an ambitious little liar. He shares this ignominy with other conservatives like Ted Cruz, but they are not being muzzled, even if they are a bit like mad dogs.
So no, America doesn't have a "social credit" problem. Hawley has a "He shit the bed and wants to sit here and polish the shit" problem. But it still smells.