The facts are that at no time could ANWR produce enough oil at any one time that would significantly influence the oil market. Furthermore, the ANWR oil will not be for the exclusive use of the U.S. but instead will be sold on the world market.
Oil is a global fungible commodity. Any additional oil reduces the tightness of supply and reduces concerns over possible shortages.
But the US lower 48 is the closest oil importing market. Sending the oil to other locations would reduce its profit with increased transportation costs. The vast majority of that would stay in the US.