>could this impact the exemption from liability for user posted content?

No, not in the slightest. Section 230 is completely orthogonal from the First Amendment. Section 230 deals with civil liability for speech that is NOT protected by the 1A and the practical effects of that as it intersects with the realities of the legal system. It has nothing to do with letting government put its thumb on the scale.

>If the social media companies claim that their choice to show or not promote or censor user content is free speech, should that speech still be exempt as if they had no control over it?

That's not what 230 is about, 230 is precisely about allowing imperfect control to be exercised. It is long, long established (rightly) that what someone allows or doesn't on their own private property and to be published under their own media is expressive, and thus protected by the 1A. If someone wishes to assert something falls outside of the 1A though (like defamation), litigating that can be expensive, as can liability if it was. There are two historic ways to handle that pre-CDA.

1. Pre-vetting (traditional publishing): With traditional media like a newspaper which published both 1st and 3rd party content (like letters to the editor), it was both necessary and practical to pre-vet everything published (and have contracts regarding liability, insurance etc). So they're responsible for what they publish, but can handle that. But that doesn't scale at all. Imagine if dang had to read every single post here on HN and evaluate whether it was legal before allowing it to go up, making edits if necessary.

2. Common carrier (classic example being carriers ala post office, UPS, or Fedex, electronic ones being ISPs): here an entity simply provides a service to anyone without any content discrimination at all beyond things like court order or provisions related to providing the service itself, like preventing anything that'd directly threaten, disrupt or damage the provider. So a mail carrier can prohibit hazardous substances requiring special handling they don't offer (or only offer with special services and costs for the equipment/training required), an ISP can deal with DoS, hack attempts and so on. But that means spam and stuff you don't want to see is passed along too.

Neither of these fit with what the people want from private forums of all flavors (from bbs to the newest social media) on the internet. It's desirable to have something that is fast and scales, where even an individual not well off person is able to have a blog and have a very low cost to run the underlying platform for that, and enable a comment section, or have a little phpBB forum or whatever else, and then let people discuss that. It's also desirable to allow there to be content discrimination (moderation, this forum is for talking about specific issues not spamming ads) that is imperfect and non-instant, without then making the hoster legally liable for anyone posting anything not protected by the 1A, and without requiring an enormously burdensome legal process for that. Section 230 was created directly as a result of Stratton Oakmont v. Prodigy, where a court found that because Prodigy had exercised editorial discretion, they were then liable for all content carried. This would destroy much of the internet. Want to have a forum devoted to cats and rockets, and so you delete posts about fish and trains? Now if someone posted something defamatory you didn't notice or realize was, you're liable for it.

So Section 230 was created to specifically to enable people to moderate content done by others, without becoming liable as a traditional publisher even if they missed things or were slow (an inevitability). They're still liable for their own speech. They're also still protected by the 1A for their own speech. At no point was it OK, with or without 230, for the government (federal or state) to put its thumb on the scale.