Yglesias hits the real motive behind the movement to create a flat tax. Surprise! It's to lower taxes on rich people. Essentially, having marginal tax rates (i.e., the thing that benefits the poor) isn't what makes the tax code complicated. It's the massive array of loopholes and credits (i.e., the thing that benefits the rich) that makes it complicated. If everyone had the same marginal rate but still had to sift through all the possible deductions, the tax code would still feel just as byzantine.
I realize that sounds conspiratorial, but it shouldn't come as a surprise to anyone who's been watching politics since, I dunno, the turn of the 20th century that the Republican party's primary constituency is the rich. Everything they do is done with the ultimate goal of freeing rich people from the fiscal and regulatory bonds of government.