I cannot offer why it is safer, but I will offer a "why" it might not be.
After I heard this story, I keep my mouth shut. Indiana is a state you don't have to tell, so I don't.
I was talking to a guy that carries and he always tells the police that he is carrying. He always gets put in cuffs while the cop clears the weapon. He said that one time, the cop took his pistol and while trying to figure out how to clear it, he had it pointing at him, the driver. He said that it made his skin crawl. It was a pistol the cop was not familiar with. That alone is a good reason to keep mum if you don't have to tell. Stuff happens.
If it was going to apparent that my weapon would get discovered, I would tell that I have a permit, but I never get in that position. First, if I keep in on me, it is concealed and there would have to be some probable cause to search me. If I have had something to drink, I keep it in the car or leave it at home (and I don't keep it in the glove box where my registration is). Again, the police need to have some kind of reason to search your car, like visibly seeing drugs or something illegal in the car, otherwise, they cannot search your car without your permission, and I don't give permission. Think of it as an extension of your house.