The "FairTax" isn't an income tax, but a sales tax. That's how it gets around the complexities of determining income.
And replaces them by the difficulties of determining a "sale". Stock sales aren't sales, for example. Neither are business-to-business sales. Only end consumer consumption is a "sale". (And it introduces vast new ranges of fraud. "Oh, we didn't eat that food. It was purchased by The Jones Household, Inc, which is a corporation and therefore not subject to tax.)
Making a progressive sales tax is harder than making a progressive income tax, because it's collected by each merchant, and there's no single record of how much you've paid so far. And the sales tax is very regressive, since poor people spend all of their money, while the wealthy make "investments". (Buying a house is a sale; buying a factory is an investment.) They combat the obvious regressiveness of it with a "prebate", a kind of guaranteed poverty-line income. (And a whole new realm of opportunities for fraud.) That means that the poor pay less. And the wealthy pay less. So to take in the same total revenue, the tax on the middle class goes up.
There are reasons to do a consumption-side tax, and it can be implemented more or less coherently with a value-added tax rather than a sales tax. (You set money aside every time you receive it, and pay it every time you spend. The net effect is that if you buy something and sell it at a higher price, the net tax is only on the profit. It applies on every single transaction, which is a lot of overhead but it eliminates a lot of meaningless distinctions.) It's still regressive, which can be fixed by a progressive income tax on high-dollar earners. I suspect that's what Gates is calling for. It's very different from the FairTax.
There are problems with that as well; there's problems with any tax system. But it's not the obvious attempt to shift the burden to the middle class, as the FairTax is.