Of course we would want that, but that's not what we get. The rich pay few taxes and the international rich pay even less, given they already have offshore tax havens by the nature of being from another country. Given a green card allows establishment of US companies which are exempt from some tariffs which are imposed on foreign companies, there's no reason to believe that giving the rich green cards will be a net gain for tax revenue. It is indeed sometimes a net loss.
They may pay a lower percentage of taxes on wealth gains than middle class or upper middle class, due to being able to delay income realization, lower capital gains taxes, and other means that are more accessible to the rich than the middle class or the poor.
But rich people still tend to pay more taxes than less rich people.
Demonstrably not true in absolute terms in many cases, or as a percentage of income in others.
Spend 15 minutes talking to an accountant on the premise that you don't want to pay any tax. If you've got enough cash, that's not a problem, anywhere in the World.
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.