The issue with flat taxing Federal incomel tax is that most state and local taxes are regressive, so you'd end up with the poor paying a much higher percentage of their income in total taxation, i.e. percentage of income paid when you add all local, state, excise and federal taxes. Right now, we're already pretty close to flat, except the very wealthy pay a way lower percentage in total taxation, especially when you add all the money hidden offshore, which we can only really estimate.
Flat tax arguments always ignore the rest of our tax system, and frame taxes as only federal income taxes, in order to fool people into supporting raising their own taxes.
The reality is, income taxes were never intended to tax the poor and middle class, they were intended to raise money for wars and defense from those with excessive income.