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.