You are mistaken. Taxing everyone at 'the same rate' equals more pain for the poor, less pain for the rich. It is absolutely unequal. You are looking at numbers and fail to grasp what the true issue is.
The ONLY fair tax is one that strives to achieve equality of burden. 15% to a poor man is a much greater burden and subsequently more painful than 15% to a rich man. A poor man's life is painful by virtue of his poverty. Because of that, Democrats feel it's fair not to increase that person's pain and to start taxation at some threshold above the poverty line.
The ONLY way to get equality of burden (in the sense of 'pain' ) is via a progressive tax. In fact, it can't ever be 'equal' in any sense of the word, but we can strive to make more equal. Always, the operative word is 'strive'. Take crime, for example, we can never eliminate it entirely, but that doesn't mean we shouldn't strive to eliminate it.
As an egalitarian, I understand the we cannot legislate equality of wealth, there will be rich people, affluent people, and poor people. No egalitarian I know would support a 'flat tax' for the above stated reason.
What we can do is strive for equality of opportunity, we can strive to close the gap between the rich and poor (though we know it's impossible to eliminate it entirely ) and strive to make the lives of those in poverty less painful.
This is what egalitarianism is all about, it's NOT about 'making everyone the same', and when the right offers that as an argument, it's a strawman.