A very key point is buried in David Lazarus' article about the flat tax proposal in the Chronicle today. He pretty much says the big disadvantages of the flat tax proposal made by Alvin Rabushka of the Hoover Institute. He points out how the flat tax would eliminate progressive taxation, but he buries the alternative by Ron Wyden and Rahm Emanuel that would keep the progressive nature of the tax code while ending the deductions and loopholes.
The flat-tax proponents have scored a linguistic coup, as they have equated tax simplification with the flat tax. A flat tax isn't inherently simpler (well, very much slightly simpler than the Wyden/Emanuel plan which would take a 10 year old with a basic calculator). But, it is a windfall for the rich.
If the proponents truly cared about the tax simplification, they wouldn't tie it to a flat tax, which is unpopular and unpassable. This makes the flat tax advocates a net negative for tax simplification. If people didn't conflate the idea of simplification (would be good, and popular) with a flat tax (bad, and unpopular) they would be more likely to support it. If the flat tax advocates truly cared about simplification, they'd back the Wyden/Emanuel plan. But, most of them have as a primary goal a tax windfall for the upper income folk.
The other big problem with the plan as proposed by Rabushka (I'm not sure about the Wyden/Ehmanuel plan) is that it exempts capital gains and dividends. The result of this would be corporations creating a special class of stock for executives who would be paid minimum wage and reap millions tax free through special dividends.
There are ways to simplify the tax code, but a flat tax is not necessary.