What effect does the Moon have on the near Earth asteroid population?
I think you answered your own question with your second paragraph. 

According to Newton's law of gravity 


the Gravitational force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.
So the biggest body in the neighborhood has the most pulling power provided it agrees with the equation.
As a consequence Jupiter and the outer planets should win out over Earth and keep them from hitting Earth or at least divert the collision course. Especially since the NEA come from their part of the cosmos.
I suppose the same logic would also work with the comets. 


Please see the following link for more:
http://earthsky.org/space/is-it-true-that-jupiter-protects-earth