The term "finished product" has no relevance as far as the FairTax is concerned. the product itself does not determine the taxation but whether or not it is purchased by the end consumer (who normally, but not always, would be an individual).
If the screws you cite were purchased by a company using them to incorporate into some sort of product to sell at retail, then certainly the screws are not taxable. Should the end consumer buy them to repair something in his home, then this would be purchase at retail by an end consumer and the screws would be taxable. "Fair" or "Unfair" doesn't enter into the matter. It is the transaction itself rather than the item involved in the transaction that is the determinant. Nor will "every item that goes into an end product" be "tabulated and taxed at the end". The thing involved is taxed only a single time with that being at the "end" in your example when sold to the homeowner for his own use.
And, NO, the consumer won't be paying "the same taxes he does today" since at present some part of the business income tax is embedded into the price of everything we buy thereby raising prices due solely to the business income tax. With the income tax eliminated, prices will decline because of that and the wage earner will have his entire paycheck to use (plus the prebate).
"Sometimes", "not always", lots of hyperbole, and guessing.