This would make quite a few Congressional hearings unconstitutional. For example: https://www.nbcnews.com/tech/tech-news/4-tech-industry-titan...
It'd probably also apply to the President saying "lock her up" about a private citizen. Is that unconstitutional pressure from an elected official?
> What does that even mean, for the US Government to have a right to speak without fear of action being taken against it by the US Government?
It means stuff like this: https://en.wikipedia.org/wiki/Speech_or_Debate_Clause