And what makes TikTok different from a newspaper, fundamentally?

Both are publishing stories written by others (reporters for the newspaper, subscribers for TikTok), and taking decisions on which stories to publish (through direct editorial control for the newspaper, through the algorithm + some direct editorial control for TikTok).

Newspapers are not platforms; they are publications. They have editors who set editorial policy. They are selctive about the content they publish and who writes it. And they are accountable for what gets published.

TikTok is mainly accused exactly of having an editorial policy, via boosting certain content that its owners prefer and de-prioritizing content that they don't. So this is a non-sequitur. And even when talking about false information, newspapers face 0 legal risks for publishing false information, unless it is defamatory (and even then, it's a civil action, no state prosecutor will investigate a paper for publishing false and defamatory information).

No one is talking about “freedom of the platform”. The literal text of the First Amendment with regard to these things is “Congress shall make no law …abridging the freedom of speech, or of the press”. If the government shuts down a venue because they don’t like what people are saying there, they are abridging those people’s speech.