The point is (also mathematically speaking) that a flat tax is less fair to the poor than having a progressive tax with a zero marginal tax for incomes below what’s required for merely surviving (i.e. no tax for people without any excess income that can be saved). So there would be room for improvement but, as well-intended as it might be, a flat tax is not making things better for the poor.

You're changing the statement and also using bad logic. The statement was "does not hurt the poor [relative to the current situation]", not "make things better for the poor", or any standard of fairness you've arbitrarily picked. Moreover your logic is flawed, because, doing things one way or the other with taxation of the rich a priori has no effect on the poor, unless you are taking into account secondary effects like price inflation, but if you are I got news for you the government is fucking the poor in those sorts of metrics in far far worse ways than marginal effects downstream of tax policy.

If you're advocating a policy of "screw the rich to help the poor", you've probably never been poor. I have and let me tell you the least of my concerns was what rich people in general were up to with their money (except for the specific rich people that were keeping me employed)