Newspapers are publishers, and are subject to rules and regulations for being publishers.
Twitter would not be mandate to open up access to everyone. Instead, the proposed legal changes would merely treat them as publishers, if they act like publishers, and therefore twitter would not have liability protections anymore.
So as soon as they make any kind of restriction on the use of their platform they become publishers? That's quite a stretch. That would have the same effect: it would require them to provide unfettered access for anyone, for any purpose.
Twitter would not be mandate to open up access to everyone. Instead, the proposed legal changes would merely treat them as publishers, if they act like publishers, and therefore twitter would not have liability protections anymore.