So when you are traveling on a public highway law enforcement has every right to search your vehicle?
Do you understand that we're talking about airplanes, and not cars? Airplanes don't travel on a highway, as a rule. Perhaps yours do, stupidpilot. This wouldn't be a great surprise, given some of your other comments.
Law enforcement most certainly does have the right to search your vehicle, given reasonable suspicion, or a number of other considerations depending on location and circumstance.
Park your car inside a secure area in which you are subject to search...and yes, your car may be searched.
Park your airplane inside the perimeter fence of an airport acting as a secure area, in which weapons are prohibited, and yes, your airplane may be searched.
So far as concealed weapon permits...most do not authorize the carriage of a firearm into places where firearms are prohibited, which include locations serving alcohol, post offices, schools, government buildings, and yes...in many cases, airports. Additionally, many airports and the authorities which operate them will prohibit weapons. Land there with a weapon on board and you're in violation.
Perhaps you're simply inexperienced enough to have had a very thorough customs inspection before...but without any permission on my part, they've gutted my airplane, smashed a liquor cabinet, and left no apology...simply because they had a question about a passenger on board. As it turned out they did search him, without his permission (because they didn't need it), and he was carrying a weapon...and he certainly was caught. As was one of his traveling companions. Not only could customs have damaged the airplane (they did...during a 135 charter flight)...they could have confiscated it.
Your "private" property can just as easily become someone else's property. Don't try waving your rights around too much, stupidpilot. They're not as broad as you might think.