Yeah, that argument takes you from regressive (everybody pays $x) to why it needs to be relative to income. Snake_Plisskin was arguing that everybody draws equal benefits, so they should pay equal taxes or else it is discrimination. My points refute that position, I think, right?
As for progressive taxation, currently we have a system that is moderately progressive up to about $1 million per year, but then sharply regressive after that. If you favor flat taxes, then you are on the "no the rich do not pay their fair share" side, since you would be arguing that we should increase their tax rates. Flat taxes are more progressive than what we have now at least with regards to the rich.
But, IMO we should go beyond that to actual progressive taxation. The chief argument is the diminishing marginal utility of wealth. Every dollar a person makes has slightly less value to them than the previous dollar. Makes sense, right? If you get $1,000 you'll spend it on the thing you need the very most. If you get another $1,000, you'll spend it on the thing you need next most. By the 100th $1,000, you're spending it on things that are far less important to you than the things you spent that first $1,000 on. Once you get up to billionaire level, each dollar has so little value that even a million wouldn't even be noticed. So, to maximize the utility of wealth, you want it to be less concentrated. That doesn't mean it should be like equally distributed or something, but you need to balance that waste due to concentration against the desire to create incentives for success and whatnot. Progressive taxation is a good way to strike that balance.
IMO that is the strongest argument for progressive taxation, but there are many. Another is that you need consumer spending to drive the economy. Ever since we started these tax policies that radically over concentrate our nation's wealth in very few pockets, our consumer spending has been growing at a lackluster pace and that prevents our economy from really taking off. Another is the obvious moral concerns with people who don't even work or do anything useful getting roughly 1/4 of all the wealth generated by all the working people while many of those people who are actually doing the work are barely able to provide for their families.