You have the philosophy right, but political reality has left that behind. While the President doesn't have any official power over the legislating process, he

You have the philosophy right, but political reality has left that behind. While the President doesn't have any official power over the legislating process, he can wield significant political pressure as de facto head of his party, via shaping public opinion from the bully pulpit and with the threat of a veto. Presidents have, for some time now, been very active in setting/driving legislative priorities.