When the top 20% owns 80% of the wealth, they should be paying 80% of the taxes.
Well a flat rate tax would accomplish this. Close loopholes, institute a flat rate tax, and bingo, you've got it.
Example:
$1,000,000 total income (just a simple number for this example).
10% tax rate (easy to work with).
So therefore, 10% of that $1,000,000 is $100,000 in tax revenue. That's the goal.
Now we split that $1,000,000 between 3 people.
Person 1 has $800,000 income, or 80% of the total amount.
Person 2 has $150,000 income, 15% of the total.
Person 3 has $50,000 income, 5% of the total.
Person 1 pays 10% tax rate, equaling $80,000.
Person 2 pays the same 10% tax rate, to the tune of $15,000.
Person 3 still pays the same 10% tax rate, which is $5,000.
All that tax revenue added up from those 3 people gives you $100,000 as we established the total should be. And Person 1, the richest guy, does indeed pay 80% of the total taxes, while making 80% of the total income.
Bingo. Easy, simple, fair. And it meshes perfectly with your requirements in your post.
This can be extrapolated to real world numbers in terms of total wealth, a fitting tax rate, the US population, and the various income levels. And it is much simpler than the current tax code which is overly confusing, screwball, and taken advantage of.
