The Act was written purposefully to skirt around the Constitution - they knew what they were doing.
But any rational analysis says that the President cannot start a war (i.e. initiate hostilities against a legitimate sovereign government) without Congress declaring it, and that he cannot have the resources to conduct any military action at all unless Congress signs the check.
However, once Congress signs the check to fund the armed forces, he is the Commander in Chief of those armed forces and can deploy them in any way he sees it necessary to execute his legal responsibilities - for example enforcing UN resolutions.
The WPA is an attempt by Congress to keep their powers while taking away some of the President's.
The “Govern and Regulate” and “Necessary and Proper” clauses of Article I, Section 8 cannot be ignored. They too apply to the armed forces just as the “Raise and Support” clause does. We have to take the Constitution as a whole.