It's an ad hominem which falls short of hypocrisy or contradiction because the soldiers and contractors sent to Iraq were not conscripted. Surely there are more substantive arguments to be made.
Cheney is defending the need for state secrets but has a proven record of benefitting from intelligence that has been obscured from the general public. Motive is extremely important here.
On several occasions. He was the Secretary of Defense for the first Gulf War, and VP when both Afghanistan and the second Gulf War started up.
This would remain a valid example of moral hazard even if Cheney had been drafted and experienced combat in Vietnam. The draft was one of the reasons the Vietnam War was so unpopular, avoiding it could be construed as an act of civil disobedience. It's not an essential part of the argument.