So let's say a person is spending $2k a month on their basic necessities - housing, healthcare, insurance, transportation, and a little bit of entertainment. And now consider two people who spend the exact same amount. One earns $30k and the other earns $90k. The person earning $90k has $66k left at the end of the year after accounting for 12 months * 2k of spending. The person earning $30k has $6k left. The $90k earner is left to pay $9k in taxes. That's 9/66 = 13.6% of his disposable income. Pretty close to the nominal 10% rate. But the person earning $30k has to pay $3k. That's 50% of his disposable income! That 10% for all isn't exactly fair, in effect.
I think this disparity is a much easier way to consider tax brackets. And I think it's a politically easier sell to both sides of the aisle, than the more abstract notions of this article. Citizen A earns $x and citizen B earns $y. They both, on average, pay the exact same share of their disposable income in tax. I think it's pretty hard to argue against that, yet the tax brackets this implies would also help inherently tackle extreme levels of inequality.