I don't see where that is BS. If everyone paid a (out of my hat) 15% tax with no deductions then everyone is paying the same percentage. That's fair.
No, since the impact on the payee is clearly disproportional.
The bottom line is that NO tax can ever possibly be fair. The two (related) issues are a) what kind of unfairness you want built into the system in order to fund public expenditure and b) how much public expenditure should be.
I would argue that the "least worst" option is to minimise public expenditure, thus minimising how much tax is required in the first place. Then in terms of raising the money, use a flat direct tax to raise part of the income in a simple & transparent way and consumption based indirect taxes to raise the balance, which is again simple & transparent, raising from each according to their expenditure. Avoid introducing any forms of allowances, exemptions or other "smoothing" modifications into the system, which will distort the system and generate unexpected/unintended turbulent sequelae.
I wouldn't argue this was fair, but to my mind it's a better balance of unfairness than either a pure direct flat tax with no indirect taxes (this seems to be your suggestion), or the progressive sliding scale direct tax with a byzantine system of allowances combined with consumption taxes (the current awful system), or indeed a pure consumption-based indirect tax (like Surefire has proposed).
I guess it boils down to how you define "fairness", which is a pretty nebulous concept if there ever was one.
I would say that rather than thinking about "fairness" in designing a tax system, it's better to consider "a minimally invasive balance of unfairness, with a focus on keeping things as simple as possible".