Scenario 1: The authorities say they are prohibiting people from marching on Wall Street streets not to suppress free speech, but simply security concerns.
The risk of their entering buildings, accosting people on the street, property damage is a problem so they can't allow the risk of violence.
However, a confidential memo from the public relations people is leaked showing the real reason was political and they used security as a justitication.
Scenario 2: an anti-abortion group posts the home address of the only abortion doctor in the state on the internet; it results in the killing of the doctor.
A memo is leaked from the group's leaders where they indicated 'if someone uses this information to attack the killer of babies, so be it.'
Depending which of these scenarios happened, it's easy to see how public opinion could be strongly for or against the 'free speech rights' involved.
But the policy needs to be decided in advance, to try to address both of them.
This is why freedom advocates say people have to expect there to be a price for freedoms - to try to prevent that creates 'police state' conditions. But as the public watches the freedoms abused and harm caused, it creates demand for stronger protections.
Whatever policy the leaders pick, they can expect to be attacked by 'the other side'.
This is why I think leaders tend to favor the position of "supporting freedoms - but dammit don't use them!"