The theory is based on this. A restaurant has asked the government to no longer be a private-only entity but to become a public entity. That's why they are called public establishments, because they have asked the government to enter into a contract with the public, the citizens of that state to serve the citizens of the state.
You can always have a private club, and serve alcohol without a license, such as kiwanis, masons, rotary, etc.
So in theory, based on this, from what I have read on the books, a public establishment has made a contract of sorts with the government and the people of that State to become a PUBLIC SERVICE provider of sorts. They have agreed to no longer be a private only establishment, but be an establishment that lets the general public in.
Since a public establishment has made that contract with the people, they are unable to say "You can't come in because you are black, white, gay, christian, so on and so on" because they are no longer PRIVATE anymore, they have already agreed to be PUBLIC and follow PUBLIC RULES.
They could turn around and become a private establishment anytime they want, and only allow members, and then no longer have to follow those rules if they want. However, since they want to engage the general public, they have decided they don't. The courts have already ruled over and over again it is illegal to prohibit any other type of discrimination, and since we are arguing that 2A is protected by.. well, 2A, would this be different?
So if the public says that you are allowed to carry a firearm, and this is a PUBLIC PLACE, it would be illegal to prevent that.