Why is anonymous free expression so important?

First, let us consider the opposite. Have you ever withheld something you wished to say, so as not to hurt the feelings of the one listening?
What I'm talking about is nothing like that. Respecting the feelings of others is a personal choice, and one I often admire. When I say the opposite of "anonymous free expression," what I mean is a sort of overt, public adhesion to what is expected.
This goes beyond considering the feelings of others. It means a public figure, at any level of power or fame, adjust their speech for any number of reasons: to avoid ridicule by colleagues; to maintain or improve position or employment; to preserve renown or even preserve their own lives; and undoubtedly there are many other motives. The point is that any cause of such action is a symptom of not being free at all.
So, if I may offer — the only truly free expression is in anonymity.
Contemporary geopolitics show that Brexit votes (GB) and Trump votes (US) far outweighed any forecast models that were offered. What does that mean? Consider, within the same classroom or work place or social space, would there be a different vote ratio for a controversial subject if the vote were cast by secret ballot as opposed to show of hands?
Even here on Steemit, where speech is still open and free, are some swayed by peers, or even by the allure of title or content that will yield upvotes?
Just food for thought. Thanks for reading :)