I addressed part of the "why" of why does the park service close the park on another thread. Roger Brown just gave a synopsis just above your post. I will just add that maybe you know how to leave no trace but most humans don't. The park services mandate is to protect the park from those people. They also have the mandate of creating access. So it could be argued that staying open is necessary for them to fulfill their mandate. But then lots of government entities are not fulfilling their mandates because Congress has shut off their funds and if they do anything but spend money on only absolutely necessary items, then they can face serious consequences.
Most people don't know how to "leave no trace". That is the simple answer.