Are you really that dense? Look at any elementary school, college or trade school.
As you observed, a flat tax and a sales tax can be made equivalent. Assuming a sales tax does not tax the consumption of education, the same result could be made by exempting teachers, professors, school administrators, school construction, etc. from taxation under a flat tax. Supporters of a flat tax are serious about not wanting to pick the winners or losers in the economy. There are no such proposals coming from the flat taxers.
Sales tax advocates are not serious about not picking winners and losers in the economy. Sales taxers are willing to place their policy preferences in the tax code. This is evident by the effort to exclude the education sector from taxation.
"Assuming a sales tax does not tax the consumption of education, the same result could be made by exempting teachers, professors, school administrators, school construction, etc. from taxation under a flat tax."
That isn't an apples to apples comparison. The FairTax does not exempt teachers, professors, etc. from paying the sales tax on their consumption. It exempts students from paying a sales tax on their tuition.
Does your flat tax proposal (and you still have not clarified which flat tax proposal you support) tax grade schools, high schools, etc?
You are grasping at straws, trying to find a flaw that does not exist.