Freedom of speech is a concept, and a legal definition in the US. It's true that Twitter has no _legal_ obligation to uphold free speech since it's not a government entity.
But if you support the _concept_ of free speech, Twitter is stiffing conversation by playing a moral judge on what is considered truth and what's considered lies.
The Constitution was written 200 years ago without any of the today's technology. Back then, all "speech" happens either live in person, or by individual printing presses. Government back then was the biggest threat to the concept of free speech, so it's indoctrinated in the constitution as a legal concept.
Today, public discussion space has moved onto social media platforms. Government is no longer the biggest threat to speech (because of the Constitution), but private companies like Facebook, Twitter, Google, etc who can just ban anyone at will and cause them to lose the ability to reach their followers. If you want to protect free speech as a concept, then we need to update our legal concept to include any platform or service that's identified as critical to public discussion.
Similar to how electricity companies are regulated as utilities companies because they're so crucial to people's daily lives, social media platforms should be regulated as speech platforms because they're so crucial to today's conversations happening in society.
This is the hard truth. You won't like it because you hate the man. But it's the truth / end devil's advocate