That's the argument, moonman. It would NOT be their focus. It SHOULD NOT be their focus. They should be focused completely on winning wars, and using every dime they can get their hands on to win wars.
That's why the VA needs to be a separate entity arguing for and spending its own money.
Vouchers, speciality hospitals, etc., those are all things that the VA could be doing regardless.
I got news for you, the military is focused on the big picture. That certainly includes weaponry and training, but there are a large number of support services that stand behind the personnel on the ground. I see no reason that the SecDef couldn't just as easily argue to get funds for both.