The parent is talking about "progressive" as in "the tax gets progressively higher as you make more money", not in the "social justice" sense. The flat tax being regressive means that the poor would have to actually pay more in taxes than they do now, which is very little. That's seen as a bad thing. Obviously there are solutions to that in flat tax proposals.
Second, "the wealthy pay very little taxes" is a meme. I mean, look at what we've got here:
http://www.pewresearch.org/fact-tank/2016/04/13/high-income-...
Those making $250K and up are responsible for 51.6% of all income tax revenue. How much more should they be expected to contribute? Remember: the truly wealthy can live anywhere in the world. There's a balancing act that needs to be established between "you're not paying your fair share" and "soak the rich."