I believe the fact the establishment has open doors and has no membership restriction for entering the establishment makes "private property" a moot case or modifies it significantly.
A private party offering public accommodations absolutely can matter, but for the most part it doesn't in this context. It certainly doesn't make the notion that private property is involved moot.
Here's the thing, if there are laws against something in particular, then there are laws against it. If there are not, then there are not. The only reason there are limitations on what can be done by an establishment that offers public accommodations is because there are laws that establish those limitations. For instance, the federal government as well as every state, as far as I'm aware, have laws that prohibit establishments that offer public accommodations from discriminating against people based on their race. There are other laws prohibiting discrimination based on sexual orientation. But in some places there aren't any laws prohibiting discrimination based on sexual orientation, so it is generally legal. There are also laws that establish requirements such as providing bathroom facilities if your business serves food, or that require handicap access infrastructure. But those requirements or limitations derive from laws, not from the Constitution placing limits on private parties.
There aren't any laws, at least around here and as far as I'm aware, that require public accommodations to allow people to carry firearms. But if there are, and when or where there aware, such would be required. The fact that carrying firearms is a constitutional right, or that it isn't prohibited by the government as a general matter, has nothing to do with it (i.e. except that it being legal in general makes it possible for an establishment to also allow it, if it were illegal more generally it might be illegal at an establishment whether that establishment wanted to allow it or not).
The point is: Yes, some place being a public accommodation does further open the door and allow governments to get their noses under the tent to make rules limiting what that public accommodation can do. But if governments choose not to have particular rules limiting that public accommodation, then that's that. Maryland could, in theory, pass a law saying retail businesses of such and such size, or whatever other conditions, have to allow people to carry firearms on their premises. But if Maryland doesn't have such a law, then those retail businesses don't have to follow such law. And Maryland not having a law against carrying firearms, or even having laws that say it is legal to carry firearms in general, is not the same thing.
Now, all that said, I would make this point. If Target has a policy against carrying firearms in its stores, that does not make it illegal for someone to carry firearms in its stores. There may be some laws that could be interpreted such that it thus became a crime, but generally speaking it would just be breaking Target's own rules. Based on that, Target could then ask someone to leave or bar them from its premises. At that point the person could be guilty of trespass if they didn't leave or if they returned. But just violating Target's policies itself wouldn't be a crime.