Does it legally matter if Dick Cheney physically uses the veto stamp at President Bush's request, but we digress?
The issue is not whether the President can delegate his right to veto legislation. The issue is not whether Congress can delegate its right to enact legislation. At issue is whether Congress can delegate the authority to enforce/enact/maintain legislation (e.g., can Congress delegate the power to enforce the $.34 stamp to bureaucrats in the Post Office, or must Congressmen themselves physically issue and verify stamps to all Americans).
You claim that since the Constitution doesn't use the word "delegate", that Congress can't transfer any level of power or authority to anyone else, especially those in another government branch.
That's simply not accurate. Although the word "delegate" is not in the Constitution, there are numerous examples of text in our Constitution in which Congress is explicitly given the right to enact all legislation necessary for the enactment and enforcement thereof. Such enforcement can include delegating power to non-Congressional bureaucrats such as Postmen, and once you realize that the language in the Constitution thusly permits said delegation of Power and authority, then the rest of your claims fall apart.
Clearly Congress CAN delegate authority to Postmen. Clearly Congress can delegate authority to the President. Likewise, the President can delegate the authority to run any part of a war to anyone he chooses.