You're wrong about Marbury v. Madison. It said, "It is emphatically the province and duty of the judicial department to say what the law is." That means that the Supreme Court is the final arbiter of the Constitution. The president is not allowed to interpret the Constitution. US v. Nixon proved that. Nixon was ordered to turn over the Watergate tapes. It didn't matter if Nixon had his own constitutional interpretation.
Of course judges have to interpret laws. They have to, in order to decide cases. But so does everyone else. When the President is sworn in, he takes an Oath to the Constitution, not to the Supreme Court's interpretation of the Constitution.
Every day the President makes decisions based on his understanding of his presidential powers under the Constitution, and most of those understanding have never been tested in court. It is absolutely the responsibility of the President to interpret the Constitution.