I seem to recall some graph indicating that even though they pay 4-5x (or higher) lower percentage of their money in taxes, the higher amounts mean they shoulder most of the tax burdon.
I don't think the percentage difference is ultimately beneficial to society, but it's not correct to say they pay nothing.
And then here, merely the top 10% of wealth holders hold 76% of the wealth, and the top 20% of wealth holders hold 85% of total wealth. The statistics are no where near what you appear to be claiming.
Anyway, I never looked very deeply into it, but I'm pretty sure that single-digit millionaires pay plenty of taxes, and I find it fair to call them "rich." I do agree that taxing income much more heavily than capital gains, property, etc. doesn't sound like a great idea.