"I see nothing wrong with Congress providing any level of oversight that they wish."
You can only have ONE commander in chief, not a 600 plus committee. The President is commander in chief, not 600 plus congressmen.
While Congress CAN withdraw funding for a war they have previously granted the authority to conduct, this sets a very bad precedent. It undermines our credibility with friends and foes.
Having Congress direct the conduct of a war is, clearly, unworkable, and unconstitutional.
I won't contest your claim that having Congress direct the conduct of a war is "unworkable". But I see no indication that such would be unConstitutional. The title "Commander-in-Chief" does not imply that the President is unaccountable.
Besides the power of the purse-strings, which has already been seen to affect conduct of the Iraq war, I believe that Congress may have oversight of general grade promotions.
From the section of the Constitution enumerating powers of Congress:"To provide for organizing, arming, and disciplining the Militia, and for governing such Part of them as may be employed in the Service of the United States,..."
In his role of Commander-in-Chief, the President is governed by Congress.