> that does not give companies and individuals the ability to choose what they host on their sites?

No, it does not. The First Amendment is silent on e.g. ISPs or payment processors blocking a particular site based on its content. Until 1897, it was unestablished whether it restricted the states in any form [1].

[1] https://en.wikipedia.org/wiki/Chicago,_Burlington_%26_Quincy...

What if I rephrase to say the first amendment does not give the government the right to tell website owners what they may or may not publish?