I think you are missing the point. If it is truly a flat tax, the number of dollars you are taxed goes up (in a directly proportional way to the number of dollars you make). So you DO pay more the more you make. It isn't even a logically arguable point.
"a flat tax actually gives a very easy way for high income people to have a lower tax rate.":
-- again that's a completely illogical statement. If it is truly a flat tax *EVERYONE* has the same rate -- that's the very definition of a flat tax.