The central problem with the FairTax on Fed Govt consumption is NOT that it taxes itself ... that is necessary to avoid the price disparity problem ... it is the fact that taxing itself generates NO NET TAX REVENUE on a portion of the tax base that is supposed to generate positive net tax revenue!
I'm sure we'll keep hearing about how "necessary" it is for the Fed Govt to tax itself; but the silence regarding the missing net tax revenue is deafening.
"... taxing itself generates NO NET TAX REVENUE on a portion of the tax base that is supposed to generate positive net tax revenue ...!"
And that is the same situation as today when the government "taxing itself" generates a net of zero also merely transferring funds from one account to another. So your observation about the price disparity problem is both correct and appreciated since that is certainly the larger problem of the two.
Shuffling of funds from one account to another is something the government is quite good at and any FairTax "paid out" is certainly returned to them within a month or so with the upshot being that the revenue involved is not a material amount in the big scheme of things.