I think the USA won in spite of the draft, not because of it. Maybe that isn't quite the correct word for it given the manpower the US could and did marshal, but patriotic enthusiasm and peer pressure was on fire in a way that would be incomprehensible in post-Vietnam America in WWII. The US owed victories on the Western Front more than anything else to industrial output, air, and artillery, unfortunately.
The Vietnam era military was notorious for having a very corrupt draft system that inevitably sent the resentful ill-birthed and connected to the Poor Bloody Infantry, but the problem extended further back than American popular memory likes to think. This guy explains it pretty well: