Twitter is merely labelling a tweet as being factually incorrect, it's not hiding the content.
I think it really depends on what you view twitter as. If it's a communications platform, like your phone, then yes 'merely labelling a tweet' is as troubling as your phone company deciding to shut your call off when you mention to a friend that you're going to vote for Biden. If Twitter is a publishing platform, then it certainly can expose its editorial bias, but one must really consider whether or not it should have to pay its writers.
The difference would be one is a company, the other a real person. No need for the company to get involved.
People who ignore the correctios and other tweets will ignore the company anyway.
Twitter saw speech they disagreed with, and they fixed it with more speech. They haven't censored any of Trump's arguments, they didn't delete his tweets. They just added their own commentary on top of them. That's what Republicans have always claimed they wanted. Argue that people are wrong, don't censor them. Don't throw people off the platform, add a fact-check.
I grew up listening to Republicans rail against the Fairness Doctrine, and I basically agreed with them on that point. Forcing private broadcasters to act like they were neutral on every issue was problematic. But now apparently that's flipped and free speech means forcing a private company not to take sides on any issue, even when taking a side doesn't require censoring or restricting anyone else's speech.
Any Republican that was genuinely anti-censorship would be cheering Twitter's move, even if they disagreed with the content of this particular fact-check.
So wave the bloody shirt, that's AOK, but say that vote by mail facilitates fraud, and you get a personalized "We Don't Think So!" message from twitter.
Twitter hosts outrage mobs that have the stated goals of getting people fired, and it has caused people I was following to quit the platform.
They simultaneously want to exercise editorial discresion while not being liable for for all the outrageous or outright wrong speech they do host.
No, it cannot even be considered a free speech issue (except insofar as Trump proposes to censor Twitter). Those of us in the con law/democratic theory community, and everyone else in the universe who is even semi-rational, use them term "counterspeech" to describe what Twitter did.
Traditionally, counterspeech is seen as the virtuous alternative to censorship---as the thing that us snotty free speech people tell those who call for their opponents to be censored to do instead. John Stuart Mill would jump up and down and pop champagne in celebration of what Twitter did.
