it turned out The Statue Of Liberty was one

Oh man. I know I complained about that exact episode in my comment, but I had actually completely forgotten that that happened. I must have blocked out how dumb it was.

"Hey, we have this monster that people love. Its two main traits are: it can't move if anyone is looking at it, and if anyone photographs it or stares at its eyes for too long, bad things happen. Also, they go around with their faces covered because they even freeze if they look at each other."

"Great, lets make one of the most famous, visible, photographed statues in the world—which has its face visible all the time—turn out to be one of them."