Over the time period mentioned in the article, the top 1%'s share of total income rose from 8% to almost 20%. If their share of income instead remained constant, reducing the top tax rates would indeed make the tax system less progressive, and an increase in the 1%'s share of income tax paid would be quite a conundrum. (You'd then need to call in other explanations, like increased closing of loopholes, to account for it.)
However, when the top 1%'s share of the total income more than doubles, it easily counteracts the lowering of their tax rates. Thus, once can claim the top 1% are paying an even greater share of total income taxes even though their rates were lowered.
While I generally agree with the idea that punitive tax rates will act as a deterrent to income (and an incentive to seek out tax havens), I think the article's "surge in taxes paid by the rich" is nearly entirely the result of the rich getting a much larger slice of the income pie.