In this case, property rights do indeed hold the trump card.
It's easy to tell who has READ the case. It was a wrongful termination case in an at-will employment state. The recent enactments of the UT legislature, including committee hearing statements, mirrored the relevant gun-rights laws' statements of no increase or decrease in existing property rights. Unlike the OK law cited above, there was no "strong" public policy found in UT law which could have trumped the employer's rights to restrict & regulate use of its leased portion of the parking lot.
The general argument holds pretty well, unfortunately, because you needed an AOL parking sticker to use that leased portion of the parking lot. The "public access" element was a bit lacking. And the Constitution is generally a restriction on the Govt's actions, not your employers. Even the ever-mighty First Amendment free speech rights tend to end at your employer's door. In the employment context, there is "strong" public policy favoring whistleblowing to expose wrongdoing.
I also believe that there is good reason to expect any "gun free" policy to be, by itself, an "undertaking" by the landowner to provide better "security" for its employees and patrons. Our "standard of care" will creep more and more towards every place being an armed/disarmed camp, with only the "special" designated guards being armed and the rest of the people there being perfect unarmed victims. Well, within the context of firearms and knives. Improvised weapons and group efforts to defeat crazy shooters or gas-leakers (think Sarin, etc.) or powder-spreaders or arsonists** will then be our best weapons. Sad, but I would hardly be defenseless even when "de-fanged". But the landowner decreeing "no guns" will face liability for attacks which are preventable under the "standard of care." Cold comfort to the dead and their survivors.
The state's public policy is embodied in its legislative enactments. Believe it or not, the UT court was and is not big on legislating from the bench...except when enacting their own internal rules on CCW and weapons in courthouses. They do tend to have a bit of an authoritarian tilt, though.
**Ever see what a single person with a Bic lighter and a can of almost any aerosol can do to an office full of paper, books, carpet and Cheetos?