I don't mean to turn this all political, but I think it helps to have a perspective on what the law is when discussing these kinds of situations.
Our investigatory system works on the basis of "reasonable suspicion." What that requires is that the police have "specific and articulable facts that would make a reasonable person think that a crime might have been committed" in order to stop you, by force if necessary, to make an investigation.
The police are not, nor have they ever been, required to determine a prosecutor's case in its totality before making a stop or an arrest, which can sometimes be quite confrontational depending on the situation. Thus, the police are not required to do things like make determinations of credibility, do a Google search (which is about as incredible as it gets), or know that there is a readily-available FAA database (which I assure you that no one outside of the aviation community is aware of).
All that I would have to do to have you stopped and investigated would be to make an anonymous phone call, saying that you stole a car and giving a description of what you looked like, where you might be found,and what the car looks like. I wouldn't even have to give my name, but if my information checked out, the police would be able to stop you.
Under our laws, that's considered "reasonable." It's a balance between: 1) total police control of everything; and 2) total anarchy. Put differently, what our system provides is that you're generally free to go about your business, unless there's a decent reason for you to be prevented from doing so on a limited basis.