regressive taxes, such as sales taxes, hit the poorest the hardest. if all one's income is spent just to eat and be housed, a 20% tax puts them out in the street.
whereas, a rich person spends a relatively small percentage of their income to eat and be housed, and much less than 20% of their income on sales taxes since they spend much less than their total income.
only works if the bottom rung has the first $50,000 of income or so is exempt, but then it is not a true flat tax.
Just tax capital gains as income at the person's tax bracket, which it is, and the revenue side would be fixed. A side effect would be that they would re-invest in american businesses to shield it from taxation, just like how it worked back in the good old days.