First, the most important part of -52 when it comes to open carry is the very first segment, "Except as otherwise provided in this article..." That "article" mentioned is Article 3 which contains all the laws from 13A-11-50 to 13A-11-85. This means that -52 must be interpreted with all the other laws in that Article which cover the subject. Every other law pertaining to carry is 50, 55, and 73-75. All of these pertain to ONLY concealed carry. -55 lists the ONLY illegal carry of a firearm is CONCEALED carry.
Second, -52 pertains to carry on "premises not your own or under your control". That is called private property. 13A-11-59 in part 4 defines "public places" or public property under the law of carrying weapons, as Article 3 is entitled "Offenses Relating To Firearms and Weapons". The first line of part 4 states, "Any place to which the general public HAS ACCESS and a RIGHT to resort for business, entertainment or other lawful purpose. If you have legal ACCESS or A RIGHT to something, that puts it "under your control". Therefore, according to the first part of -52 and part 4 of -59, you have a right to carry open or concealed in/on public places as defined by law.
You have access and a right to public places. This makes "public places" premises under your control to conduct business, for entertainment, and any other legal purpose as defined by law. You can meet your friends in the parking lots or in the restaurants. You have access and a right to park in the parking spaces, walk on the sidewalks, meet and talk with friends, put food or merchandise into your shopping cart, etc. So, according to the law, open carry in public places is legal because they are under your "limited" control, not just on your own premises.