No offense, but your question is faulty. State v. Kerner is not the binding case on this matter, therefore how they relate to Kerner is irrelevant. Kerner is a 1921 case and there have been been several other cases on point (though I do not have the cites).
The gist of current case law (as I understand it), is that municipalities may not outright ban open carry in public, but they can regulate it, i.e. minimum weapon size, a la Chapel Hill.
Municipalities may also ban ALL wepaons from municipal buildings, parks, etc. but must do so by passing an ordinance and posting the prohibition at all entrances (just like any other property owner who wished to ban CC).
Furthermore, even wth a ban, it is okay to have the weapon in your car (i.e. carrying, arrive at park, put weapon in trunk while enjoying the fresh air, reholster upon leaving)