Back to basics:
A "fair share" system of taxation would take the same amount of money from every person living within the borders of the U.S., and subject to its laws. Take the total required to operate the government, divide by the population, and assess everyone their "fair share." It is the same in any sane club or association. Figure out how much you need to get by, set the dues accordingly, and collect the dues. If you don't pay the dues, you don't get ANY benefits from the organization. What could be more fair than that?
Parenthetically, one might note somewhere in this discussion that an income tax is prohibited by the U.S. Constitution, and was only made possible by Amendment - which was sold on a number of lies. Ideally, in the minds of the Founding Fathers, the national government could generate all of the revenue it needed by tariffs and fees. But I digress...
Assuming that a tax on incomes (rather than on wealth) is the most viable option, a less "fair" system - but more workable - is one that taxes all incomes at a fixed percentage: the so-called "flat tax." It has the built-in advantage of minimizing the taxes on those with the least ability to pay, and maximizing taxes on those who can most "afford" it. But it is inherently unfair because it punishes financial success with higher payments to the government.
You will find very few organizations that fund themselves on such an unfair basis. Indeed, who would want to join such an organization? A masochist?
An extremely "unfair" system that is even more inequitable than the Flat Tax is a graduated income tax, where not only do the successful pay more because they earn more, but the rate of extraction is graduated so that their "contribution" goes up exponentially as their financial success increases. This goes beyond "unfair" to the level of diabolical. It truly punishes success and discourages innovation, creativity, and productivity.
The only "good" thing one can say about the American graduated tax system is that it has been worse than it is now, in the not so distant past. Also, thankfully, we do not have quite the crushing burden of the European welfare states, so the top rate here is less than it is there.
The proof of the unfairness of the graduated tax is the sheer size of the Internal Revenue Code and its associated Regulations. Much of the "fluff" in the IRC and IRR is comprised of elaborate (and not so elaborate) mechanisms by which various constituencies can avoid paying taxes - legally - by doing the things that Congress wants them to do. Build things, invest in things, grow things, don't grow other things, make your stuff more energy-efficient, and so on.
If the graduated income tax were actually "fair," then none of this would be necessary. You would simply pay the tax and be done with it. But it isn't fair, so Congress has to manipulate the rules to soften the impact.
And at the other end of the spectrum, we have Congress trying to buy the votes of the unsuccessful and the wretched by minimizing or eliminating their obligation to pay taxes, or - believe it or not - causing the government to PAY THEM NEGATIVE TAXES through the perversely-named Earned Income Tax Credit!
The one universal principle that is always ignored is this: EVERYBODY should pay something. Whether it is a payroll deduction, a percentage of a person's profits in business, or a deduction from a Social Security or welfare deposit, every emancipated adult in the country should feel a pinch of taxation, so that they can be vested in the business of Government. We have fully half of our (*******) population that pays no Federal income tax each year, and THIS is a crime.
The fairest tax: A flat percentage tax with a fixed maximum amount - whether its a hundred thousand dollars or a million, I don't care. But there should be an amount over which the government has maxed out its share of a person's income.
