a flat tax is naturally progressive
This is flat out error and wrong. All taxes are regressive, even the ones that "target" the rich. Remember the "Luxury Tax" during the Clinton era? It was short lived because it didn't collect any revenue and cost thousands of jobs or regular people as the rich suddenly stopped buying Luxury Items (boats, planes, limos etc). It didn't hurt the rich, it hurt the people who made stuff for the rich.
It was so bad, so quickly that it was repealed in very short order. You don't hear about it, because it doesn't fit the "Progressive" mantra of "Progressive" taxes.
The fact of the matter is, taxes are a necessary evil, because they are regressive, either intentionally (Cigarette Taxes), or unintentionally (Medical Device Taxes under Obama Care).
Demonize the rich all you want, I don't care. But the moment you try to take from them via "Taxes" you're not going to hurt them at all, you're gonna hurt those that work for the rich. That's the thing about "disposable income", they don't have to use it at all or they can use it to keep more of their own money and avoid taxes, and that always hurts those whose incomes depend on the rich spending that money.