Not that I am saying that austerity itself solves anything, but the real reason austerity did not crush the US when we had sequestration but hurts most of Europe is monetary policy, The ECB's laser focus on low average inflation makes the countries that are doing worse overall suffer in case of austerity, because it makes NGDP drop. All you have to do is look at the charts. Since the ECB did not do any QE, all you have to do is plot GDP and ECB rates. In the US, unconventional monetary policy kept going up when things started to look worse (if we skip the first year of the crisis or so, when the Fed was rather passive)
So it's not that austerity works, but that you can have austerity and work if you have a competent central bank. If you switched central bankers but kept the same government spending, The US is the one in double and triple dip recessions, while the EU would be doing better.