in 2014 the top 1% of income earners earned 20.6% of all income and paid 39.5% of income taxes collected.
the bottom 50% earned 11.2% of income and paid 3.45% of income taxes.
https://files.taxfoundation.org/20170201091804/TaxFoundation...
[1] https://www.washingtonpost.com/news/wonk/wp/2012/09/19/heres...
Even with a moderately regressive income tax the rich will be paying more total tax even though they are paying a smaller percentage of their income as tax. Therefore you can not make any conclusions about the fairness of a tax by saying that the top 20% of earners pay 80% of the total amount of income tax (it would have been absolutely crazy if this was not the case, unless everybody had the same income).
As such, I do not see how this article can be used to say anything about whether the tax is progressive or regressive (and as you mentioned, whether it is fair altogether).
Edit: I suppose if you figure in sales taxes into overall tax burden you could call some states regressive. But that is not a feature of their income tax rates.
https://www.marketwatch.com/story/45-of-americans-pay-no-fed...