To be clear, Overturf wasn't so much about him being in a public place as it was about the exemption he claimed.
You as the property owner can only LOC if your property is not considered a "public place", an unfenced front yard, is considered a "public place" via overtutrf. Therefore LOC in incorporated area on your front (unfenced) lawn is a 12031 violation.
The court in Overturf found that the legislature's use of the term "have" implied only "possession", and that possession is distinctly different than "carrying". The court then defines "carrying" as transporting from one place to another.
Nothing in this section shall prevent any person engaged in any lawful business, including a nonprofit organization, or any officer, employee, or agent authorized by that person for lawful purposes connected with that business, from having a loaded firearm within the person's place of business, or any person in lawful possession of private property from having a loaded firearm on that property.
So, if you are immobile, or your firearm is resting on a nearby table/chair/whatever, then you're covered under the above exemption... maybe.
However, as we learned in Theseus' 626.9 case, the trial court may get kinda crazy when determining what "private property" is...