It would also never work because a blanket flat sales tax is horrifically regressive and hurts the poor far far worse than the rich.
There's nothing remotely "fair" about the fair tax, and this has been explained in considerable detail elsewhere (and isn't really appropriate for this thread- but the short version is- The poor spend ~100% of their income, so they'd be taxed on ~100% of it.
The rich spend a much smaller %, which is why they have $ left over- and would generally have even MORE of it left over under such a system.
And to make up for the amount lost by not taxing high incomes the rate of the flat rate would be pretty high too.
Every national-sales-tax scheme that's at all practical and useful (VATs in many countries for example) only work when you have either (or a combo of):
A litany of exceptions of things that aren't subject to that tax (or are taxed in a more complex manner)
or
A rebate provided back to lower income people to make up for them being taxed far more heavily
Further- all such countries have a much, much, stronger social safety net assisting low income folks than any we're willing to pay for here in the US.
I'm not even saying such a system isn't better than ours-in many ways it is, in others there's arguments it's not-- but it's never (in any working system anyway) nearly as simple as you suggest.
And as you can see in the EU, Canada, and elsewhere- even folks who DO have such a system still ALSO have to layer more complexity on top because governments need tools to encourage or discourage certain things.... for example encouraging home ownership, or encouraging the purchase of electric vehicles, or encouraging having children.