A modern steel SCUBA tank is typically pressurized to 200 bar, or about 3000 psi, making it 200x more useful than a bag. An inflatable ball is only inflated to about 0.5 bar/8 psig, human lungs are only capable of about 0.1 bar/2 psi. This would be about as useless underwater as one of those little one-liter bottles with integrated mouthpiece that scams show being inflated with a bicycle pump...
Still cool to see swimming like this depicted in 800BC! Puritan attitudes towards swimwear as well as imperial navies conscription of non-swimming sailors seem to have caused us to imagine swimming as a modern activity that we do better than anyone in history, but I expect that people throughout history did lots of swimming.
I also find it interesting that he's depicted wearing a helmet, I wonder if that's to identify that "this guy's an Assyrian foot soldier" or if they actually swam in their helmets!