Just a data point for the above argument.
I was rich for a few years, defined by a family income exceeding $200K per year. However, I did not pay any income taxes because I formed a corporation that operated at a loss. The trick is to know how to operate at a loss- a trade secret of the rich- taught to me by an unscrupulous auditor.
I do not know how many rich people are incorporated. But I suspect that a very large number of them take advantage of such loop holes and pay taxes at an effective rate that is much lower than even what the poor pay- what I pay now.
I would be in favor of a flat tax rate if it were to include corporations and corporations could be kept as honest as individual tax payers. But tax dishonesty pervades the US corporate culture. For example, two sets of books are often kept, one for the IRS showing artificial losses, and one for the SEC showing artificial profits.
Apparently God fearing morality does not penetrate corporate life. My guess is that the tax structure actually promotes secular immorality as no individual becomes responsible for dishonesty. The responsibility lies with corporate policy.
In my case everything I reported was moral in the sense tht the IRS approved it. But to use Mike's exprssion, it was a house of cards. One of the pillars in that house was my job as a scientist, which fueled the corporation. I made investments into the corporation. When I lost that job, I then lost my house, had to declare bankruptcy to avoid 0.5 million taxes on the house, and eventually lost my family. So in the end, I never paid taxes on all the money I made when rich. The rich do that all the time. It's the middle class of salaried workers that carries this nation. But the rich that run it to their advantage.