That is a very good observation. It is all too convenient to claim that one supports the troops, while opposing whatever it is that they do. Convenient, yes, but also disingenuous. As you indicate, this helps to expose the claim for the lie that it is.
As for the Democrats' sulking vote to deny funding for the troops, it may turn out to be a blessing in disguise. For it tends to make plain (to any centrist and independent observers who take notice of such things) that the Democrats are not just anti-war, or even pathologically anti-Bush. They are also anti-military.
And this is an attitude that does not please many people anywhere to the right of, oh, the folks at MoveOn.org. Or perhaps the Daily Kos.