That's not really true. 99.9% of the tax system is progressive[1] and the middle (14% effective tax rate on 59k annual) and upper middle (17.5% on 95k) classes pay less than the highest quintile (27% on 260k). Even the top 1% (35% on 1,570k) pays fairly reasonably, although capital gains eats into that- it would be closer to 45% if it was taxed at the same rate.
The real problem is the .1%, the people who's annual income is in the hundreds of millions. These are athletes/actors and "businessmen" who are really just investors in their own company, and they form two distinct taxpayers. The athletes/actors pay rates ranging from 20% to 35%. The ultrarich businessmen are heavily clustered around 10-20%. So heavily that both groups together paid an average rate of 16.62%[2] in 2007. That's just... incredibly insane.
I think part of the opposition to fixing this problem comes from framing it as a middle class issue. It's not even a middle class + rich issue, it's an issue for even the extremely rich. It's not an "us vs. CEOs" issue. It's "everybody vs a subset of the richest thousand people". There isn't an income level above which you are the enemy, the enemy is all of the people who skirt by paying taxes. Long term capital gains taxes are too low and ultrarich tax loopholes are abused liberally.
[1]: see page 35 of this CBO report- https://www.cbo.gov/sites/default/files/114th-congress-2015-...
[2]: page 10 of this IRS report: https://www.irs.gov/pub/irs-soi/07intop400.pdf
For some perspective, "the 1%" starts at about $350k annual salary (think middle-of-the-pack surgeon.) At that level, you're paying a lot of taxes.
There's an excellent recent Econtalk on taxation in the US.
How does any of this make sense?
That's simply tax fraud, it's completely different from the other examples you're calling out..
Would a flat-tax fix this? In theory: but if we set that "flat tax" to 15-17%; (and simply pocket the cost savings from simplification of tax code) - that seems like a win. But the very poorest quintile can't pay that. (and are equipped to dodge that, by simply not working and collecting aid or taking black-market work - cash-basis labor, or selling drugs, etc.) - the top quintile will also have the means to avoid paying their fair share. As always.
It's those in the middle, who have to work REAL jobs, and can't hide their income though either black-market means, or tax-shelters: which still work in a flat-tax system.
The government will need to raise overall rates to compensate for this, and the middle gets screwed even more.
That's an IDEAL system.
In the REAL world: the wealthy will still lobby for special deductions.
Personally, I'd like to see the mortgage deduction go away for non-primary residences.
Stuff like "dancing-horses" deductions and the absolute murder "small businesses" get away with, (like; taking their personal Truck as a deduction, by calling it a "work truck") - that can be horse-traded around. But I think there is the most to be gained from the mortgage deduction on non-primary residences. This is probably America's most sacred cow. And therefore, is most likely politically impossible. This is why we've been at a stalemate over tax-reform since 1980.