It's interesting, because usually that lobbying is for something that generally helps people and can sometimes be seen as good legislation (such as tobacco regulation).
In this case, however, only harm is the outcome of removing the strengths provided by the first amendment.
Debatable on either front but pretty tangible in either case. It's not even clear what the first amendment means in this context because it can be evoked on either side, the freedom of private platforms to moderate as they see fit or the freedoms of users to speak on essential infrastructure.
Thus, all of these censor loving platforms should be held accountable.
This is the consequence of what you're proposing, a politically right or left leaning website say, simply by virtue of exercising freedom of association, would automatically become liable for content.
It seems pretty clear that this is unworkable and would, instantly, destroy the entire public internet.
If a website moderates per pre-defined and agreed upon terms, then they should not be liable. If a website is going to moderate with a left-lean or right-lean, it should be up front about this agenda instead of selectively censoring or flagging comments.
The courts have already ruled that even with right of way the government has no right to content standards on cable - now it magically is going to apply to non-excluding websites? Utter nonsense.