Heroic Inventor stories have always been popular and can stay out there for a long time. Whitney was quite a showman, and he does seem to have performed some tricks with some lock parts to stave off criticism of his tardiness in filling a contract. He had entered into musket-making for the government as a way to finance his cotton gin venture, the patent of which was under legal assault, but had had no experience in their production, and consequently did not deliver according to schedule. But, in any case, in the end he didn't make his contract guns with interchangeable parts. That would have required a great deal of tooling- jigs, fixtures, machine tools- that he did not possess. It was something that was of great interest, however: Thomas Jefferson had brought the concept back from France, after visiting French armories ( where it was being promoted / explored by H. Blanc) and had become enthusiastic for it. So Whitney would have known that his little show in front of Jefferson and his government would be very well received.

It would first be actually done for some years in the John Hall shop at the Harper's Ferry Armory beginning in 1817, making his breechloading rifles. It would gradually take over as the main goal of armory production at Springfield in the 1830's, 40's, and become known as The American System. Though it did have many virtues and drove American precision machining to new heights ( the US would supply milling machines to England in the 1850's, to make their Enfield rifles), it would also become a barrier to innovation; it's hard to scrap tons and tons of specialized jigs, fixtures, and tooling and make tons more for a new design. As a result the Trapdoor rifle continued to be made long after it was obsolete. And, similarly, Henry Ford would discover the same problem when, in 1927, he tried to shift a factory totally devoted to mass production of Model T automobiles to the brand-new Model A, and had to close his factory for months to switch over..

Two very good sources on the origins of the American System are David A Hounshell's From the American System to Mass Production, and Merritt Roe Smith's classic Harper's Ferry Armory and the New Technology.

A very good article on the whole story of Whitney's arms making is The Legend of Eli Whitney and Interchangeable Parts by Robert S. Woodbury. Technology and Culture, Vol. 1, No. 3 (Summer, 1960), pp. 235-253