First off, I think all of us have learned the unfortunate downsides to a libertarian approach to free speech over the past 20 years. Any free speech purist would do well to remember what their legal recourse would be if somebody made elaborate false-allegations about you that harmed your career and marriage - we have many legal tools that protect us from harmful lies. It is not remarkable that there are people who believe in "free speech" but do not consider disinformation about more nebulous bodies that don't have standing for a lawsuit (eg defaming vaccines in general, epidemiology, democracy, ethnicity, climate science, trans people, etc) to be something that should fall under that protection, any more than threats or perjury fall under that protection.
Why is it actionable when you make dangerous public lies that hurt somebody's pocketbook, but not public health?
Secondly, the US constitution says "shall make no law". What law was made here? What legal action was taken? There wasn't even a threat of legal action.
Government workers should be free to contact private organizations and speak to them freely and make requests of them. "The government would like this content taken down for public good" is not making a law, it's making a request. It's making their opinion known, and government functionaries are allowed to have professional opinions. Something like "In my professional opinion as a public health worker, this content is dangerous advice that will get people killed, and in the interest of public safety it would be best if readers were protected from it." That is a reasonable thing for a government-employed professional to do and say.
That said, I think they crossed the line here when it became a demand instead of a request. When the government starts ordering people around instead of just making the public interest known, it can easily be argued there's implied threats there.
IANAL, but I'm assuming in the end that's where this will land during the appeals - that sweeping injunctions against various government bodies communicating with social media companies will be lifted, but the court will find against the government on this case.