That might be the wrong way to go about it, but I sympathize with the sentiment. If you pretend to be a public square you must act like a public square.
Now many here already raised an obvious objection - someone tending to a small blog should be able to police it. That is sensible. So I suppose we need to find a good way to draw a line between a private club and a public square.
Even bigger problem might be your own content hosted on public squares. I wonder if Instagram/Fb/TikTok are declared a public square, does it mean that I can't delete someone's comment under my own photo, status or video because it violates their free speech even though it is my profile or page?
I would think it should be covered by the same doctrine.
Just as an exercise. Suppose a private club is where all members are know to the owner and all associate members are know to a member. A public square is a place where people can stumble into without any direct affiliation (search, directory, discovery, recommendations, shares).
Under this definition it matters not who is hosting your pictures, it only matters if strangers are welcome. And if they are then all strangers are welcome.
Now my idea here does cover the hypothetical theater - you want people to discover it, but you don’t want them causing noise during the performance. So it’s not a great definition. But it serves to illustrate.
As it happens the US Supreme Court is also sympathetic - https://en.m.wikipedia.org/wiki/Marsh_v._Alabama
Now many here already raised an obvious objection - someone tending to a small blog should be able to police it. That is sensible. So I suppose we need to find a good way to draw a line between a private club and a public square.