Well, the basic version is that it wouldn't raise anywhere near enough money. Especially if it was only 10-15%. According to the article, Carsons plan would increase the deficit by about $10 trillion dollars over ten years. The article doesn't say, but for it to be revenue neutral, it would need to be a much higher percentage of income.
So, $10 trillion dollars is a lot of money. But how much is it relative to the size of the annual budget? The federal budget is about $3.8 trillion a year (call if $4 trillion to make the math easy). If you want to cut taxes so much AND keep the deficit under control, you'd have to cut spending. No two ways about it, simple math. So where do you cut $1 trillion a year?
All discretionary spending is only about $1.1 trillion a year. So you can either slash literally all discretionary spending (goodbye military, science funding, border control, etc.) or you can touch entitlements, primarily Social Security and Medicare. Just wait for the howls when you suggest cutting Medicare benefits.
In short, simple math dictates that if you decrease revenues so much via a tax cut, you either increase the deficit or cut spending. If you try and cut spending by that much, you'd have to touch programs that are incredibly politically popular. It's simply not at at realistic that you'd be able to pull it off. There are too many vested interests who depend on federal money. Are big defense contractors going to take kindly to losing billions in annual fees? What about Tea Party types? How are they going to react to shutting the Border Patrol?
And all of this ignores the effect that it would have on lower class consumption. Right now, the Republicans are correct: the poor don't pay a lot of federal income tax (ignore sales taxes and the like which are highly regressive). Why's that? They don't have a lot of money. It's the same reason people rob banks; that's where the money is. If there's no money, there's nothing to tax. So at the lower end, you pay basically nothing in federal income tax and it ramps up relatively slowly. If, all of a sudden, you make it a flat tax instead of a progressive tax, you deeply impact people at the lower end of the income spectrum, i.e. those with the highest marginal propensity to consume (MPC).
What's MPC? I basically means how much are you going to spend of the last dollar that you earn. Put another way, if you make $10,000 a year you're going to spend every last dollar, because you have to do that simply to live. Rent, food, transportation, etc. If you make $100,000 a year, you'll be able to put some away and that very last dollar that you earn, you might only spend 50 cents. Some aspects of life are really just fixed costs (everyone pays the same for gas and McDonalds). Once you cover those fixed costs, you can set aside more for discretionary spending and savings. If all of a sudden your tax burden jumps, you're much less able to cover those costs. Quite simply, a flat tax from the first dollar would be devastating to poor people.