The Angels can do so many things. They can bend time, climb inside
your mind, hide in pictures, steal your voice, mess with your
perception, leak stone from your eye… New York in 1938 was a nest of
Angels and the people barely more than farm animals. The abattoir of
the lonely assassins!
In those terrible days, in that conquered city, you saw and understood
only what the Angels allowed, so Liberty could move and hunt as it
wished, in the blink of an eye, unseen by the lowly creatures upon
which it preyed. Also, it tiptoed.
Adding onto this, there is at least some in-story precedence. In "The Time of Angels", when the Doctor realizes that all of the statues on Alfava Metraxis are, in fact, Weeping Angels, he suggests that he and his party may not have noticed them because the Angels could have been generating a perception filter, although he also considers that he and his party may have instead just been "thick." However, as the Weeping Angel on the video clip managed to create a deadlock seal out of nothing earlier on in that same story, it seems entirely within the realm of possibility that Weeping Angels, especially ones as well fed (and therefore more powerful, as also elaborated in "The Time of Angels" - DOCTOR: "Dinner to an Angel. The longer we leave it there, the stronger it will grow.") as the ones in Manhattan, who had a perpetually-generating food supply, could have created a high-level perception filter that would have put peoples' attention off of the Statue of Liberty. This was actually lightly alluded to in "The Angels Take Manhattan", when Grayle said, "These things are all over, but people don't seem to notice."