In fact, the progessivity of the tax system has very little impact on inequality. You can have a highly progressive system (like the US, which by some measures has the most progressive tax system of any first world country), or a relatively regressive system like most of Europe, and it just doesn't matter. In which case, anything to simplify the tax system in the US is probably a good idea.
Further, the "rich" in your example are able to easily evade most taxes. Consumption taxes - specifically if implemented as a VAT, as in Europe - are effectively impossible to dodge, even if you're very rich. You may not pay a lot as a percentage of your total income, but you will pay. The (nominally) highly progressive system in the US is unable to guarantee that.
(And in any case, as recursive pointed out, the prebate check would make the system arbitrarily progressive, undermining your entire complaint. Although personally, I'd favour just implementing a straight up European-style regressive VAT, and if you want to redistribute income, do it via welfare payments; that's what they're there for. And unlike tinkering with the tax code, they actually are capable of effecting significant redistribution. Your entire argument seems predicated on the idea that the US tax code has something to offer that the typical European tax code does not. Will all due respect, I question this. The US tax code is a disaster, and US prosperity has been achieved in spite of it, not because of it.)