Yep. One of the first issues you'll encounter when learning to dive is getting your body to sink (that's what all those lead weights are for). I also don't think an inflatable goatskin has enough air capacity to be relevant.
One cubic foot of air displaces one cubic foot of water (compression notwithstanding). A cubic foot of water weighs 62lbs. I think you're vastly underestimating the bouyancy of an inflated bag. Try to pull a standard birthday party balloon underwater. It's nearly impossible to do. You would need a tremendous amount of extra weight to submerge a goatskin full of air.
I was referring to the bladder's capacity to store breathable air. I'm well aware how buoyant an inflated bag (or a wetsuit, for that matter) can be :)
Did you see the second image -- how about if it was deflated and used as a re-breather and was principally to stop exhaled bubbles from giving your position away to archers as you crossed a river/moat?