That incorrect perception is what the law is all about. But the LEGAL question is simple to answer. Do police have to get a warrant or permission to search a person's vehicle?? The answer is yes.
We are not talking about police here but private property owners. They would have every right to require that anyone wishing to use their parking lot and work in their business sign a consent to search such vehilce at any time, all are part of their efforts to enforce their no gun ban. Just as one’s right to free speech doesn’t prevent them from being fired if they mouth off to their employer.
Let's perfect that analogy. You need the permission of the state to drive that car on the public road (operator's license, state inspection certificate, mandatory insurance, etc.). That permission has to be applied for. The state has requirements that you have to meet in order to be granted permission. The state asks you questions and requires you to prove things to them. There is no reason I see that a private company can not require you to get similar permission from them before granting you access to their land. Nobody has an inalienable right to be on another person's land.