I haven't really heard anyone talking about a flat tax, like a flat dollar amount, on income taxes -- although there may be some, and I simply missed it.
Nobody's talking about a flat tax or a flat wealth tax; I put those into show the large spectrum of things that you can potentially flatly tax. Most people hear the term "flat" and think of it only in terms of income as though that would be the default one, even though it's just one of many things to potentially tax. (The existence of income tax in this country doesn't go as far back as the founding of the country, for example.)
Whenever people use the term "flat tax", they are usually talking about a flat tax on income. But this is arbitrarily asserted as the fair thing to tax flatly. Why not wealth? Because currently, we're just used to income taxes.
One variant is the proposed "Fairtax", supported by libertarian Gary Johnson and some other fiscal conservatives, which is a flat consumption tax (a federal sales tax, basically) that is made into a slightly progressive consumption tax with a prebate. It would be less progressive in terms of income taxation than the current system, and highly regressive in terms of tax on wealth.
I have heard discussion about a flat tax rate. What do you think about a flat tax rate, like 10% or 15%?
While that is not progressive, it still does not produce the same kind of extreme difference as shows up in your example.
With a flat tax rate of 15%, a person making $10,000 pays $1,500 (provided there is no allowance in the tax code for basic living expenses) and a person making a billion dollars pays $150,000,000.00.
A flat tax rate on income does produce exactly the extreme difference as I showed in my example, because it would be a
regressive tax on wealth. In other words, on average, the more wealth you have, the smaller percentage of your wealth you'd be asked to pay in taxes each year.
In your example, a person making $1 billion in income would almost certainly have a financial net worth of billions of dollars, while a person making $10k would likely have a zero or negative financial net worth. So one person would be paying more money in taxes than their entire net worth, while the other person would be paying a tiny fraction of their large net worth in taxes.
Even with the currently somewhat progressive tax on income in the U.S. (with loopholes for millionaires, primarily in the form of low dividend and capital gains taxes), tax on wealth is currently regressive. So, right now, the bottom 50% of people are paying a larger share of their wealth in taxes each year than the top 1% are paying as a percentage of their wealth in taxes each year. Even using myself as an example, I paid a smaller percentage of my wealth in taxes in the last year than I did the year before that, as my income and wealth have grown.
The reason is that people on the lower half of the income spectrum have little ability to build wealth, since their income goes primarily towards the basics of living and they're taxed on it. People on the higher half of the income spectrum have disposable income that they can use to compound and grow into larger and larger wealth. So we have a situation where the entire bottom 50% of the population collectively owns only 3% of the wealth, and the top 1% has over 30% of it. Since we're taxing wealth regressively.
A flat income tax would be far more regressive than even the current system (and the current system has led America to have among the highest of wealth and income inequalities in the developed world), which is why developed countries do not do it. The only way to turn a flat income tax into a tax that isn't regressive on wealth would be to have a very large tax credit: like 40% of all income over $100k is taxed. But that would be a progressive income tax disguised as a "flat" tax.