"Granting public access into private property doesn't have to be a binary "all or nothing" sort of thing."
But that is, in fact, exactly how you're approaching it.
I don't see this aspect of the law being addressed, either...
For the moment, and ONLY for the sake of this argument, I'll temporarily stipulate to the point that you hold, that "private" property conveys absolute rights.
What this law does is allows employees to keep firearms locked in their cars.
Those cars can also be construed to be "private" property -- that, of course, varies from state to state, I don't know how Florida looks at it.
But, what right does the employer have to say that the employee is forbidden from keeping a firearm in their private property?
So, in this situation, whose private property rights then become paramount?
The business owner? Business owners do not have a generally recognized blanket right to search the insides of privately owned vehicles, even if they are parked on the employer's property.
Then there's also the sticky wicket of what "private property" rights exist in situations where the facility is not owned by the company, but leased? Whose private property rights are paramount then?
Sorry, I'm with Florida on this one. A enumerated right in the Constitution should, in a balancing act between two groups, take precedence over a "traditional" right, especially if that traditional right isn't extensible to non-residential buildings.
But, let's for a second, take a look at a few of your examples:
The DOD contractor, for example. Restrictions such as those are not a matter of rights, there a matter of national security. Saying a gun locked in a car is a threat to national security is a stretch.
The PETA protesters? Protesters do not have a right to disrupt the operations of a business. It's a stretch to say that a gun locked in a car is going to disrupt business operations.
The public golf course? A golf course will generally only deny service to individuals who are dressed inappropriately in terms of safety or to protect the integrity of the grounds. They're not going to let someone play who's wearing ice skates, for example.
The bar owner certainly can ask patrons not to smoke inside the bar. But we're not talking about a gun owner brining his gun INSIDE the business. Locked in the car, remember?
"Regardless, who are we to tell him he can't?" And conversely, who is he to interfere with the second amendment rights of otherwise law-abiding citizens, especially when their terminus point is parking lot at the job site, and upwards 90%, or more, of their travels between the offices and point of origin are not work related?
It's equally absurd to think that simply because you have a business that you have a blanket "right" to interfere with the exercise of another's Constitutionally protected right, but not quite as absurd to think that 1,000 years of English Common Law and 200+ years of Constitutional law regarding private residences somehow magically morphs over to cover businesses.