What matters far more is that Eurozone adopted tight money policy, while the US didn't. People love to focus on Fiscal policy and even many economist got drawn into that. But the reality is that monetary policy actually sets aggregate demand.
Doing fiscal austerity isn't bad if your monetary policy keeps AD stable. Doing the opposite and spending more is also not really bad as long as monetary policy is correct.
The US had bad monetary policy that was partly responsible for the recession of 2008/2009 but corrected it eventually. The Eurozone never corrected in and held to an extreme tight monetary policy for years and at the same time doing austerity.
And that simply can't work, if your AD is collapsing, trying to save enough to pay back huge nominal debt is simply impossible.
Yanis Varoufakis who was Finance Minister in Greek proposed debt payments indexed to NGDP, so that if AD was going down they would have to have less debt payment. But this was rejected by
Countries with their own currency, like Sweden, Switzerland and so on did better despite having many of the same pressures and not going on fiscal spending spree.
I think it does make some sense for the government to invest in infrastructure during a recession. But not because of the Keynesian thinking of pushing up AD, but rather because it just make sense invest in infrastructure and its slightly cheaper during a recession.