Realistically, Eichengreen's argument is considered so solid these days that even fringe nutjobs with an anti-FDR or anti-Keynes agenda don't tackle it straight on. Instead, the typical plan is to insinuate that fiscal stimulus doesn't work because the US did not achieve full recovery before WWII.
Of course, anyone who has taken even an undergraduate course in US economic history should know that fiscal contractions on the state-level throughout the mid-1930s basically offset expansionary fiscal policy on the federal level (the seminal paper is E. Cary Brown's "Fiscal Policy in the Thirties" [1956]). This is why no-one in mainstream economics believes fiscal policy was truly expansionary in the US during the 1930s. That said, if you're asking why this is relevant that is a good point, since the important thing about leaving the gold standard is not that it led to immediate INFLATION so much as that it stopped DEFLATION and began the slow process of recovery. And this is the observation made by the piece and it is absolutely correct: once the United States left the gold standard the country's near double-digit annual deflation stopped and its process of recovery - however slow and faltering - began.