If you have an issue with how Twitter, FaceBook, YouTube, et al. moderate content, you are completely free to create your own Internet service which you may moderate as you see fit.
The US government exists to insure domestic tranquility, provide for the common defense, and promote the general welfare. The government is not interested in protecting the rights of citizens. Instead, it is the rights of citizens which are meant to protect us from the government. Other laws may dictate how businesses are allowed operate but the first amendment absolutely does not apply to private businesses.
These were private companies infringing on rights.
The president sent in the national guard.
The civil rights act did not make up any rights - it created law to enable enforcement and protection of rights people already had, but which were being infringed - both by states and by private businesses.
The social media monopolies are infringing on our rights. There may not as yet be a legal remedy for that, but it seems likely that there will be, as in many ways these companies are more powerful and wield more influence over people than governments do.
> The social media monopolies are infringing on our rights
The right to free speech is not a natural right; nor does it apply to anything but freedom from government persecution on the basis of speech. Freedom of speech is a right that is granted by the constitution and has been further defined by case law over the past 250 years. The Supreme Court has pointed out that private companies are not subject to protections enumerated in the constitution, which, again, only apply to the government[3].
[1]: https://legal-dictionary.thefreedictionary.com/Right
[2]: https://en.wikipedia.org/wiki/Natural_rights_and_legal_right...
[3]: https://en.wikipedia.org/wiki/Manhattan_Community_Access_Cor...
Once upon a time, the rights of citizens only protected them from the federal government's actions. States could (and did) infringe on these rights. Corporations were much less powerful at this time, so were not much of a concern.
We ended up fighting a war over this, and one result was the 14th amendment, which binds the states to also protect these important rights.
Now a days, corporations are powerful enough to infringe these rights. We've dealt with this in the past when "trust busting" was more of a thing. There is established precedent for not allowing monopolies to infringe on people's rights.
There's nothing special about the "private sector" that allows big corporations to infringe people's rights without due process.