The legal right to free speech is indeed only about the government. Free and open discourse as a virtue for citizens of a republic is a much broader concept.
And even the founding philosophers of the freedom of speech concept said there were limits to the concept - i.e. JSM said that there is a harm principle, where if the expression causes harm then the liberty to express it should rightfully and forcefully be admonished.
Its simple. Google did not violate free speech in terms of law. But freedom of speech is much more than just a law, its also a social responsibility that we can choose to care about or not.
Personally I think americans should choose to defend freedom of speech, even if they aren't legally required to.
This is one that really misses the mark. The First Amendment of the United States Constitution says that the government can't persecute you for your speech, but Free Speech is a separate principle that says that we should allow a free forum for people to put out their thoughts.
It is in california, among other places.
Federally, there are about 5 or 6 regulations covering workplace speech, the most sweeping of which is usually considered to be the NLRA.
If you google for nlra and lexology, you can see a number of articles about the scope.
There are actually several US states that have political activities as a protected class. But you are correct for most US states, that is, employers in those states are free to say "All Democrats - you're fired!" and it would be legal.