Defamation, for example, is not protected by the First Amendment, and deciding whether some information is true or not is crucial to determining whether it constitutes defamation.

This is irrelevant - there are all kinds of speech that are generally accepted to be outside the scope of free speech. Political speech that includes false claims does not fall under that and never has. If taking down "false speech" was something the US government can easily enforce, we'd make China look like a shining beacon of freedom. And the proper recourse to such speech is the court, not poorly paid and overworked social media moderators.

Also, it's important to keep in mind that the Trump ad in question does not make claims that are easily proven to be false - they are merely BS claims that are entirely unsubstantiated. I don't think they'd be taken down even if a rule against false claims existed and was properly followed. There's no evidence given for such claims, but not all claims without evidence are false.