He understands of course twitter is a private company, but he thinks (my opinion) that twitter reached a status where it is more than a company, it became de facto the public square for people public discourse. I would include other places alongside such as reddit and facebook, reddit is the worst. In being a "public square" it has the responsibility to allow free speech, to allow ideas to be discussed and let public decide which ones are good or not, and not a board of people deciding "this can be published, this cannot".
There are multiple problems with that line of reasoning.
First off, why should a discussion board or website lose their right to free association simply because they got popular enough? The state would be the only entity capable of enforcing that, and why should the state have that power? I totally get giving the state the power to prohibit businesses from pumping toxic waste in the ocean or engaging in wage theft, but telling a website that it can't kick someone off of their property for using racial slurs or spreading dangerous conspiracies is not something I think the state should be doing. While private property rights are not absolute, restrictions on their use should be narrowly-defined and have a very, very good reason for existing (e.g., preventing direct harm to others).
Second, the law by necessity frequently deals in arbitrary limits, and using popularity to determine a line above which a website is stripped of their right free association would be one of those arbitrary things. How will popularity be measured? Total active users? Average daily post counts? Once we've determined what we'll be using as our criteria, what's the threshold? 100,000? A million? Ten million? Why that threshold and not some other threshold? What if it eventually gets lowered to "one" and therefore no website can have a code of conduct governing user behavior?
If a site or board cannot have a code of conduct in their TOS, that would effectively force every board to allow just about anything that wasn't illegal, their moderators largely toothless. Every one of them could have the potential to become like 4-Chan's /pol/ board. Website owners need the ability to have rules dictating what constitutes unacceptable behavior on their site in order to foster a reasonably healthy and productive community, rather than one that's just some anything-goes cesspit that drives away all but the worst sorts of people.
Giving governments the power to limit freedom of association like how some are proposing opens up a huge can of worms and could potentially set a dangerous precedent.