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Who is going to bring them to court?

Who is going to pay for these court cases?

Who has the money to draw them our indefinitely?

Companies are already responsible for what they post.(see Dominion lawsuit)

Companies aren't, and should not be, responsible for what other people post on their websites.(See Section 230 prehistory)

Should they be responsible for what they algorithimcally show to users?

No. And you literally wrote why... Algorithmically promoting something isn't a matter of an opinion expressed by the company.

But neither is a company liable just because they engage in moderation.

You want to make the civil court system responsible for litigating the truth or falsehood of every comment, post and claim made on the internet?

Why not just make a separate court system instead of tying up the existing one? We couldn't call it the Ministry of Truth because it's the US, but maybe Bureau of Truth?