Deductions/credits/etc are where the rubber meets the road, though. I don't think you can just waive your hands and say "give certain people vast deductions." That's exactly where all the complexity and equity issues are going to arise, flat tax or progressive bracket system. What are the deductions? Are they capped? Who gets them? Is there an income threshold? Do they phase out? If so, at what income level? Would a credit be better mechanism? If so, should the credit be refundable?
Simplicity and fairness are opposing forces in the Code. The simplest income tax is not fair. The fairest income tax is not simple. It's a sliding scale and reasonable minds can differ on where we should aim for on that scale.
And if you really want to take on the ultra-wealthy and the low effective tax rates many are paying, you will likely need to address the realization requirement, capital vs ordinary, and tax free step up in basis at death (among other things). A simple flat tax, at least as you've described so far, doesn't address those things.