The Gate being literally surrounded by the mountain, with no magical means of getting through the mountain, makes a lot more sense than it being connected to the statue.
Having it in the statue would be like if Girard had left huge defenses in his “random patch of desert” and then left the Windy Canyon gate completely undefended. Girard’s double-bluff made sense because it told people (falsely) “the gate isn’t here” after they had already made their way past all the defenses.