You state "The mark that the group seems comfortable with is 23%. So, at the register, the price tag says $100, your final bill (assuming there's no state sales tax) will be $123." This is incorrect. The final bill would be $129.87. The 23% comes from the fact that $29.87 is 23% of $129.87. This is known as the tax inclusive rate. And the FairTax is unique in using it for sales taxes.
To quote the FairTax FAQ (#47):
"I know the FairTax rate is 23 percent when compared to current income and Social Security rate quotes. What is the rate of the sales tax at the retail counter? 30 percent."
They claim they do it this way because that's the way income taxes are quoted.
Going back and re-reading the FAQ linked above, Tony is correct. I'm not necessarily going to agree with him regarding his suspicion of malice on behalf of the Fair Tax camp with this. I got the inkling from Tony's email tone that he's not entirely on board with the proposal. But I appreciate him writing in and pointing out the error none-the-less.
All that said, in the example that I provide there is still substantial savings to the consumer at a 30% rate. Fair Taxers should just state that from the outset, since the retail counter rate is the one that has most impact on consumers, and thusly, have the most effect on the measures success.