Sure, the top 20% pay 95% of the taxes.
But what the story DOES NOT MENTION is that the same top 20% owns 95% of the nation's wealth.

You couldn't get the rest of the quintiles to pay any more than they do, because the other four quintiles don't own any of the wealth. They can't pay any more, because they don't have any money.

So, yeah, the rich pay all the taxes if only because the rich literally own all the money.