You're right, it would tax the underground economy.
The current income tax system gives the economic advantage to the cheater and the crook.
It would "tax the underground economy" as effectively as Prohibition rid America of alcohol and it would eliminate tax crime just as effectively as Prohibition eliminated untaxed moonshine.
In order to cheat and leave no paper trail under the income tax system, you have to be a partner in crime with your employer or be self employed. The risk of getting caught is so high that most employers in legitimate businesses won't go there even if they are self employed.
In order to cheat and leave no paper trail under the Fair Tax system, you merely need to be a partner in crime with one of the tens of thousands of black market sellers that will be selling stolen goods out of their car trunks or selling untaxed services for cash.
For the Average Joe, a 23% savings will be a huge incentive to tax cheat by purchasing goods and services on the black market. For the average crook, the huge demand for untaxed goods will be a equally huge incentive to engage in the theft of such goods.