However there is another interpretation which says that ability to remove someone from your private business is also an expression of free speech. And that private companies as well as individuals are free to choose who they do business with. But there is one nuance here - this is better served under another principle which is "freedom of association". Freedom of association states that you can refuse to deal with anyone you don't like. However in USA this principle recently lost its value after certain group of Christians were not allowed to refuse baking a cake for a gay couple. So it seems there are certain protections.
But then there is a third thing which is that in some countries (and I am not sure here about USA) the president has the right to send a message using any media channel he sees fit, and the media channel cannot refuse. Informing and communicating with the public is one of president's duties and refusing to send his message infringes upon it. If USA has such a law and if Twitter could be interpreted as a media platform then it could be against the law for Twitter to do what it is currently doing. Unlikely thou, as twitter has a team of lawers and likely they were consulted beforehand.
> Zooming all the way out I don't see why Twitter really care about such a fight
Companies are ran by people and those people take political sides. Twitter is on the democratic party side, and the election season is coming. I predict we will se a bigger coordinated effort spanning Google, Youtube, and Twitter, and some other big corporations as we get closer to USA president election date.