OK, I think I realized the fundamental problem with that article: the assumption that the purpose of tax policy is to acquire an amount of money from a person and not to, say, influence the purchasing power of a dollar.
If in 20 years goods have become more expensive as a result of one tax policy and not the other, there is a significant difference in the behavior of this hypothetical person, and that's a significant part of answering which policy we should choose. They may not change total consumption but they may spend significantly more money on the same basic things everyone else is spending on, and less on luxury goods/services. Or they may consume more to maintain their standard of living, and save less. (Or they may decide whether to consume more as a result of one policy or the other.)
High taxes on income at the highest bracket have the effect of reducing the number of people with a certain high income, and thereby reducing the number of people able to participate in a market for very-high-priced things. This causes downward market pressure in the price of such things - for instance, the number of people willing to pay millions for housing in a city like SF or NYC will go down, so it limits the bubbles in those cities. It meanwhile does not affect people below those brackets, which in a sense is equivalent to scaling up their incomes (and the price of goods) while leaving the rich alone.
I (and many people here, I would guess) work in an industry where our own salaries have gone up significantly, and housing prices in our hubs have gone up significantly, and e.g. minimum wage hasn't gone up proportionally. As a result we negotiate for higher compensation so we can afford housing, mostly to maintain our purchasing power and not to increase it—or in other words we're effectively devaluing the dollar as this happens. This is a problem taxes could solve, and many of us think it's a legitimate use of government taxation power. If we get taxed in a way that depresses the after-tax incomes of everyone who's competing for the same housing we are, and as a result housing is cheaper, I think we're going to notice the increased tax out of proportion to the actual tax revenue.
Consumption taxes would maybe solve this for the rental market but wouldn't be as helpful for sales; if anything we'd rush to buy even more quickly before prices go up and before we'd have to pay higher consumption taxes. I realize I'm picking a very specific example here but I think investments like real estate are what people expect the "fat cats" to be using most of their wealth on. If we reduce the gap in purchasing power between the paycheck-to-paycheck taxpayers and the fat cats, then we've achieved the "logically impossible" goal in that post. You may not be boosting consumption per se, but you are changing the prices of goods and making better products/services available to the non-rich for the same amount of consumption.
Of course none of this is convincing to the author of the post, who readily admits that "brilliant" economists disagree with him but is setting his own simplified understanding of the world.