A really nice video on proportional vs. progressive taxes-- and the often-unethical powers exerted within democracy. But it ignores the far larger burden of FICA taxes on income. One other notable omission: a "flat" tax with exempted income is also progressive-- in that those with higher incomes pay higher average tax rates.
The video is also nice/helpful in providing examples on wage inequality vs. income inequality vs. wealth inequality. It makes the point that any of those measures are merely proxies for well-being. And it rightfully implies that one should not be treated favorably by the tax code if they decide to engage in more leisure or more pleasant job characteristics than others.
One big omission in this arena: there's nothing on variance in household types. A big chunk of measured inequality at the household level (a big deal these days) is a function of whether people get married, stay together, etc.