I mean, the US saves very little money and yet was able to stand up to VW Group, so I'm not sure what "has to have savings" has to do with your example.

Obama's observation was that you can't run a trade surplus with Southern Europe for decades and then get all surprised when they run out of money to pay you with. In a similar vein, you can't have EU cross-border banks without a EU cross-border insurance deposit scheme, because then one bank can potentially drown a small country, which is what happened during the Eurozone crisis.