Part of the fight is the question are these outlets platforms or publishers. If they are a platform anybody can say anything and the people running are not responsible for what is said. If they are a publisher they are (or in my opinion should be) responsible for the content, meaning they have to verify everything and can be sued if somebody is putting out misinformation.
The problem (in my opinion) is that the outlets are trying to have it both ways. They want to screen posts for political content they don't like, but don't want to be sued for something written by somebody totally unconnected with the website. Laws like this are an attempt to solve the problem. The social media has to publish whatever gets submitted, and accepts the protection from lawsuits.
The alternative is they put a out a clear and detailed statement of what is and is not allowed, and employ an army of fact checkers to examine ever single message for compliance. If something gets through they can be sued.
Edited to add: the first amendment protection of speech has always applied to the publisher. If I'm publishing a newspaper the government can't tell me what I can or can not put in my newspaper. If you submit an article and I decline to print I have not violated your rights. You can start your own newspaper anytime you want, and publish what you want.