You have misunderstood the meaning of the word "public", because it is often used for government provided things. Many private establishments are public spheres, the most common example are Pubs - short for Public Houses.
So you have completely misunderstood the word and arguing from your mistaken assumption.
I understand that it is conventional to speak of some places that are private property as being public, but they are not public in the sense that actually counts for the protection of free speech.
Other examples of public spheres in private hands are the internet provider you're using to read my comment and write your own reply. The ISP doesn't interfere with your free speech. E-Mail is another great example of a public sphere. You can freely send your opinion to any and all the people you want.
In some countries, all unexploited nature is considered public space. Trump's Twitter was deemed a public forum.
It has nothing to do with government ownership, even though most physical public forums are government owned, such as parks, streets and squares.