At least he does on many key economic issues. In his blogpost today he comes out explicitly for breaking up the big banks, for an immigration policy that is focused on bringing down the wages of high-end wage earners (I'm going to put words in his mouth and assume that he means doctors and lawyers and not lower paid STEM workers), and for reducing protections like copyright (I'll read in patents also) that benefit special interests.
He also commits himself to a monetary policy that is considerably more expansionary that what the Fed has been pursuing. Whether it would have done the trick in bringing the economy back to full employment is an open question, but for those of us who care about poor people, the difference between an economy with 4-5 percent unemployment and the current economy will swamp anything we can hope to accomplish with food stamps, TANF, or any other politically plausible government programs.
Anyhow, I don't know how many troops Douthat has, but if there are many conservatives who actually want to beat up big banks, drug companies, doctors and other professionals who get rich through protectionism, and to seriously push for full employment policies (maybe Douthat would go for a lower-valued dollar), then on economic issues progressives might have more allies among conservatives than in the Obama administration.