A humorous way to express the same idea is that the non-rich are actually sponging off the rich:
"The average American household of 2.64 people receives almost $13,000 worth of federal benefits, services, and protection per annum. These people would have to have a family income of $53,700 to pay as much in taxes as they get in goodies... Only 4.8 percent of the population -- 12,228,000 people --
file income tax returns showing more than $50,000 in adjusted gross income. Ninety-five percent of Americans are on the mooch." -- P. J. O'Rourke [1]
The data you link to shows that the top 5% income people pay 58.55% of all taxes, while the the top 50% pay 97.22%, as you indicated.
So as a simple-language summary we might all agree that the rich pay a majority of the taxes, while the richer half of the country pays nearly all the taxes.
Note that the "richer half" starts at the median income. For the US, the median income is about $52k. That isn't particularly wealthy. When the bar for "nearly all the wealth" is set at that point, it is not surprising at all.
"The average American household of 2.64 people receives almost $13,000 worth of federal benefits, services, and protection per annum. These people would have to have a family income of $53,700 to pay as much in taxes as they get in goodies... Only 4.8 percent of the population -- 12,228,000 people -- file income tax returns showing more than $50,000 in adjusted gross income. Ninety-five percent of Americans are on the mooch." -- P. J. O'Rourke [1]
[1] https://books.google.com/books?isbn=1555847153