Are the retailers and producers of the goods that the illegal immigrants buy not already paying income tax on those transactions?
I mean, if an illegal alien buys a Coke, Kroger, Coca-Cola, their employees and shareholders pay income tax on the transaction today. Under a sales tax, Kroger will collect and remit the sales tax. Your example makes no sense; the taxes are paid either way.
You are correct that some taxes are paid today in this scenario due to corporate income tax costs being passed on to the consumer. However, the under-the-counter worker is failing to pay income and payroll taxes now. Under an NRST, the entire tax burden falls on the consumer at a single time -- there is no way to evade only part of the tax. Since illegal immigrants aren't eligible for the FCA, this means that their tax burden is the NRST rate on their retail purchases. In essense, they will have a higher effective tax rate than a legal resident.