The idea that working is something I -could- do, to make money to better my life, rather than working is something I NEED to do, so that I don't starve or die from exposure, is very appealing.
Come to Europe - it's like that over here! Mostly, at least.
Imo, flat taxes are perfectly fine when combined with a guaranteed minimum income, which is in and of itself highly progressive. It pretty effectively emulates a strongly progressive tax structure without any of the overhead and much simpler logistics.
I come from a family that is by no means poor, and I believe my parents are in the highest or second-highest tax bracket; they can very well afford to pay that much. Some friends of mine (who are also well above the poverty/government assistance line) couldn't, though.
Even with a minimum income, a flat tax rate would be a great injustice.
It's wrong to automatically assume that the flat-tax rate in such a situation will be the current lower tax rates. That's the goal of the conservative version of the flat tax rate, but it's not the only way to implement it.
The sustainable idea is to set the flat tax rate at a fairly high rate (around what your parents are paying), and give equal payments to everyone. This emulates a lower tax-rate on the less affluent, and the existing high tax rate on the rich, but it does not require tax returns, asset testing, auditing. You could also bundle payments for family benefits into this system (which would increase to the adult level, and go into personal accounts when you hit 18).
e.g. the flat-tax rate could be 40% of income, but everyone gets, say, $10000 per annum, which effectively replaces welfare payments for the very poor, and tax returns/progressive tax breaks for the middle class. You can look at existing tax figures and work out what level to set everything at.
Effectively it would "end" welfare by paying the same cheque to everyone in the community, and since you can't be "kicked off" the base payments for working, there's no disincentive to take a job, or "bracket creep".