Laws vary from state to state, and sometimes even between jurisdictions within a state. In Nebraska for example, yes I can get the permit, but the law governing that permit says that any private property owner may prohibit persons from carrying on their property by simply posting this sign: https://statepatrol.nebraska.gov/vimages/shared/vnews/storie... . If I want to be a law abiding gun owner I have to respect that (despite the fact that it is silly), otherwise I could have the permit revoked, and possibly face other legal repercussions.

It falls under the same umbrella as trespassing. You have permission to remain on private property within set rules.

From a legal point of view, it’s no different to “no shirt no service”.

(yes, even with a permit. I have a government-issued license to operate a couple of classes of motor vehicle. You're perfectly entitled to forbid me from doing so on your property.)

I once worked at a place where a few of the guys organized an informal "bring your gun to work day" each week, where we'd go target shooting after work. Well at some point corporate got wind of it and, of course, freaked out and banned it. We had enough people who objected to the ban to make a stink of it (I hesitate to say we were "up in arms over it"), and eventually after negotiation we agreed to keep everything locked up in our cars in the parking lot.