Freedom of speech doesn't mean that you have vocal cords. It means you are free from reprisal for your speech. This was a government appointment, and the story says that the DOE confirmed that the reason for his removal was because of his writings. So, the DOE punished him, by removing him from this appointment, based on his writings.
Put in a concrete, extreme example, consider if all teaching positions across the US required a statement of belief (say, in god). This would clearly be a violation of the establishment clause (really, tied to freedom of speech, but you could make another example that is more closely linked to the current situation). However, if some private catholic school required its teachers to believe in god, that would not be a violation, because the constitution binds the government, not private individuals or groups.
So, in a clear answer to your question, "why should it in this case?", it should because the constitution is a set of rules for the government to act upon, not for private companies.
Hence the historical importance of tenure for judges, for example. I assume his university position is likewise protected by academic tenure?
So I respectfully disagree. Whatever this is, it isn't a freedom of speech issue.
Sounds like people on this team are too busy googling each other instead of shit like "can super glue gum up our oil well?"