There are two different kinds of speech at issue in the Supreme Court case.
1. The first kind of speech is the posts themselves, the contents of which were written by users. For the social media site, this first kind is third-party speech. For the users, this first kind is first-party speech.
2. The second kind of speech is what the social media website does with the post (boosts, downranks, deletes, marks with tags, bans the user of, etc.). For the company, this second kind is first-party speech. (Less relevantly, the second kind of speech is also anything a worker at the social media company writes on the site in official representation of the social media company's views.)
Section 230 declares that the social media site cannot be held liable for the first kind of speech. Social media sites can still be held liable for the second kind of speech. Holding the social media company liable for boosting a harmful but First-Amendment-protected post would violate the social media company's First Amendment right to moderate. The right to moderate comes from the First Amendment, not from Section 230. My rule of thumb (not always applicable) is that harm which wouldn't have happened if the post content had been different is actually harm caused by the first kind of speech, and therefore liability should rest solely on the author of the post, even if the social media site boosted the post.
Suppose that I make a post about eating disorders on social media. (Posts discussing eating disorders are protected speech.) The social media site boosts my post. Some kid sees it and later develops an eating disorder (correlation, with the question of causation to be decided in court). The kid's parent (or caretaker) sues the social media site and argues that the social media site should be liable because the social media site boosted the post.
Scenario 1. If Section 230 didn't exist, then the social media company would have to go through the entire court process. The social media site argues that "First, social media websites have a First Amendment right to moderate. Second, our moderators could not be expected to foresee that a mere discussion of eating disorders would cause more harm than help. Third, the liability should fall on the user who posted the speech." (Depending on the scenario there might also be a fourth argument such as "Our algorithm made the recommendation. Since we didn't knowingly boost a harmful post then we cannot be held liable for the post." This is the "knowledge" issue that applies to just about any third-party liability case.) The social media company loses a lot of money, but the court rules that the social media company was not liable for the post.
Scenario 2. Since Section 230 does exist, then the social media company can argue that "This lawsuit attempts to hold someone online liable for distributing speech made by someone else. Section 230 says that this kind of third-party liability cannot exist [except for the exceptions: federal crimes, 'intellectual property', and electronic privacy]." (In this scenario, the "knowledge" issue is irrelevant and doesn't need to be brought up in court.) The court declares that the social media company cannot be held liable for the post, and dismisses the case early.
Either way, the social media company would not be liable. But Section 230 is still necessary to prevent social media companies from being overwhelmed with having to go through entire court cases, especially if either party brings up the "knowledge" issue. Regardless, the parent could sue me, since I authored the post. I would be the only appropriate party, if any, for the parent to sue.
(There's no guarantee that the court would actually find meaningful harm or liability from the particular post. And obviously, there could be no third-party liability on the social media website if the court in either scenario were to find no first-party liability on me.)