My point is that the illegal nanny who doesn't pay income taxes today will not collect and remit sales taxes under the so-called Fair Tax. The economic contributions of illegal aliens cannot be captured by a sales tax. No illegal activity can be taxed under either system.
When an illegal purchases something (say a loaf of bread), that is a legal transaction. The income tax captures the sale of the loaf of bread and the sales tax will capture the sale of bread. Legal activities will be captured under either system.
It doesn't matter which system you choose, a sales tax and an income tax capture the same exact transactions.
You're the one who is mistaken (not incoherent) since the discussion was never about the nanny not paying income tax suddenly turning around and collecting and forwarding sales tax to the state. By definition - as I originally said - THAT WILL NOT HAPPEN.
When that nanny spends the money for retail items, though, the FairTax DOES collect a substantial tax revenue rather than none at all under the income tax (since, remember, she doesn't report the income). Not reporting the income under the present system is the equivalent of not collecting/forwarding the sales tax ... they are both examples of illegal activity which by definition is not taxed.
It is when the money from illegal activity is spent that it is "exposed" to taxes but under the income tax system as I have said there is almost no tax revenue generated compared to that generalted under the FairTax.
Your "capturing" the sale is nonsense since on a $100 sale the income tax will only capture the tax on the profit from the sale. If the retailer has a 15% margin his profit would be $15.00 and at a tax rate of 25% the tax revenue generated would be $3.75. Under the FairTax, the tax revenue would be $23.00. There's a massive difference in the amount that is "captured" - and that's the heart of the discussion. The fact that both may pay some amount of tax is meaningless; it is the amount that is the main consideration.
You are, simply, misinformed.