Secondly, I'm not a big fan of how you (and quite a few economists) conflate personal and corporate income taxes. It really muddies the water as even questions like "what even is income" is different between the two.
Additionally, flat tax doesn't go well with the left. On the corporate end, exemptions in a lot of ways are the carrot to the sticks of fines and penalties that we use to regulate business (encouraging positive behavior). On the personal end, you greatly increase the tax burden on those least able to bear it, with an increase in benefits being totally orthogonal (most flat tax proposals intend to be revenue neutral). Someone making $25k nearly doubles their tax burden with a 15% flat tax.