@NomNomNom:
The original flat fee approach was recently attempted in the UK, as the "Poll Tax". Poll taxes have a very, very bad reputation.
The problem with flat Poll taxes is easy to work out if you consider the issue of "disposable income". For the sake of argument, let's assume there are no other deductions, such as National Insurance:
Taxpayer 1 makes £20K / year. Poll tax is a flat fee of £1000. That leaves Taxpayer 1 with £19K after tax. Subtract the costs of living: a car (say £100 / month, including fuel), a mortgage (say £800 / month), two kids (say £500 for food, clothes, etc.) That leaves Taxpayer 1 with just £180 or so per month as disposable income to spend on anything else, including his savings.
Taxpayer 2 makes £200K / year. Poll tax is a flat fee of £1000. That leaves Taxpayer 2 with £199K after tax. Subtract living expenses: an expensive car (say £250 / month, including fuel; we'll assume he's into Chelsea Tractors); the mortgage will likely be quite high (say £2.5K / month). His kids, Quentin and Quentinella are spoiled rotten, so let's make that £10K / month on their private school fees, food, fancy clothes, etc.. That still leaves Taxpayer 2 with over £3900 per month as disposable income to put into savings and other luxuries.
Taxpayer 3 makes £2 million / year. Poll tax is a flat fee of £1000. That leaves Taxpayer 3 with £1999000 after tax. Subtract living expenses: a very expensive car (say £2500 / month, including fuel; we'll assume it's a mid-range Ferrari as this isn't a billionaire we're talking about here) ; an expensive house in a posh part of London: say £10K / month. His two kids, Kensington and Chelsea, will be asking for the same toys and fancy schooling as Taxpayer 2's, but XBox360s and a flat screen TV will cost the same, as will the private schooling, so that's another £10K or so per month, but let's add a live-in nanny as well: £2166 / month. That still leaves Taxpayer 3 with £141,000 / month. Or, to put it another way: he still has about £1.9 million to play with each year. More than enough for that yacht he's had his eye on.
(You can assume that the income is combined across two parents if you prefer, or you can assume all three examples are "single parent fathers". It's the total income that matters in any case.)
As you can see, the wealthier you are, the less you have to worry about living costs, taxes, etc, and the more money you have to spend. Once you reach billionaire levels of money, your basic living expenses become little more than a rounding error.
A flat fee may be the most logical solution, as you're really just paying for a service provided by the government, but it is perceived as being "unfair" because those at the poorer end of the scale feel like they're paying more as a proportion of their income than the rich. And they'd be right.
The thing is, as a proportion of their income, the rich are paying proportionately bugger all for pretty much everything, so this isn't a particularly compelling argument when viewed objectively. As you've pointed out, nothing else is charged according to your earnings, so why should the services provided by a large organisation (like a national government) do so? The wealthy can afford to pay for their own private healthcare, private pension schemes, private education and so on, so they're hardly a burden on the State; why should they be required to subsidise poorer citizens too?
Nobody wants to be a billionaire because they like staring at the Queen of England's, or Benjamin Franklin's, face. They aspire to wealth because it means they never, ever, have to worry about where the next meal is coming from for either themselves, or their family.
Proportional (i.e. percentage of income) taxation really is fundamentally Socialist. It exists—and is tolerated—because it feels "fair", but it's also a lot more convenient and easier than fixing the many other problems that make it effectively impossible to impose a flat annual fee instead. I can think of a few solutions, but they'd require some serious socio-cultural changes. (E.g. making public transport free at the point of use. And that's just a small part of it.)