In describing your question, you have come pretty close to answering it.

The design of a early gun can be summarized as lock, stock, and barrel. The lock was the firing mechanism. The barrel was the tube that contained the explosion and guided the bullet. The stock was the wooden handle that all the other pieces were assembled into, to hold them. In order to make a lot of anything, in the pre-industrial world ( if there was a big market), craftspeople specialized. To make enough guns for an army, there were specialist gunsmiths working in big shops, either working in an armory or supplying one. In the case of 18th c. England, the specialists making locks and barrels were mostly in Birmingham. The lock making shops would have smiths using forging dies to generate rough pieces, and other men with a variety of files doing the shaping, others polishing. Barrels would be forge welded by smiths , bored using water powered boring machines, then ground to profile by polishers using large water powered grindstones. Those metal parts would be sent to the Tower in London, where barrels were certified to be the right caliber, proofed to make sure they would not blow up, and the locks certified. Then those metals parts would be supplied to other specialist shops who would assemble them into a stock.( Yes, this is simplified- the assembly would be a part of the stocks shaping process, and there were brass pieces, called furniture, and triggers.... but you get the idea.)

This specialization was good enough for the purpose of making a muzzle-loader. The only important uniformity was the caliber, and everything else, like the stocks, could just be pretty close- patterns were lent to the various makers, and the Brown Bess was called the Land Pattern Musket. However, the French ( precisely, a gunsmith named Honoré Blanc) realized that if they could have all the parts of a musket interchangeable, all exactly alike, it would be possible to easily repair them in the field. They didn't get very far in this- that meant working by hand, with all sort of measuring gauges and patterns, to make identical things to very tight tolerances, and this was much more work. However, the idea was immensely appealing to the US ambassador to France, Thomas Jefferson, and he brought it home, and it took root in the minds of American armorers.

In the early 1800's, a gunsmith named John Hall came up with a breech-loading rifle , and got a contract to build them for the US government. He was able to set up his shop at the Harper's Ferry Armory in 1819 in order to do so- where many specialist barrel-welders, lock filers etc were already making guns ( and many of them were sons, grandsons of gunsmiths in the area- so they had some very good hand skills). Breech-loaders had been around before then, as you notice. But a muzzle-loader only had a barrel with one open end- so containing the 10,000 PSI pressure of the explosion of the powder charge was relatively simple. As the barrel of a breech-loader was open at both ends, the breech mechanism to contain that high pressure gas had to fit pretty tightly. That mechanism therefore had to be carefully filed and fitted, which took a lot of time. Which is why breech-loaders were quite expensive.

Hall wanted to make his breechloading guns with interchangeable parts. He also wanted to make them relatively quickly, in order to make a profit. And he knew that machine metalworking tools were being pioneered by the British. So, he set up his shop with milling machines and lathes- but, more importantly, he set it up also with gauges, jigs and fixtures. That made it possible to do repeated operations: to cut the side profile of, say, one left breech hinge piece, then set another in the same milling machine fixture and cut another left breech hinge piece, and get the same dimension. Now , it would be far too much to say that suddenly there was no point to having guys with files, grindstones: they would stay very important. But a lot of the work was accomplished.

Hall's shop failed, for various reasons . But Hall's system was noticed, and copied. As breech-loading designs evolved, requiring more and more complex mechanisms, the system was crucial. The Springfield Armory adopted it- you could even say, it became their religion. One very large shop in Vermont, the Robbins and Lawrence shop, became a source of special metal cutting machinery and tooling, and making , yes, breech-loading guns: the Sharps rifle would mostly be made there. In the later 1800's, the hundreds of small gunsmith shops that had been making muzzle-loading rifles and guns had mostly given up, and a gunsmith became mostly a repairman.

Of course, the same can be said for blacksmiths making hinges, locksmiths making locks: they, too, stopped manufacturing and became more often repairmen. The ability to do repeated operations was useful for more than guns- it quickly was adopted for manufacturing everything: sewing machines, bicycles, whatever. It's just that the job of making enough guns to outfit an army had required so much production capacity, it's understandable that it would be what's called the American System really first started. Again, this is somewhat simplified: there were other places where some mass production techniques were being used earlier, like Matthew Boulton's factory in Birmingham. But Hall's innovation really was key.

Even with mass production techniques, equipping a big army would always be a challenge. When the Dreyse rifle appeared, many countries, like Britain , France and Belgium, settled initially for just a conversion of their existing muzzle-loaders: the Snider Enfield required a new breech mechanism, but the rest of it was a 1855 Enfield rifle. And even the US: despite Hall's design, it avoided the expense of converting the armories over to breech-loading guns through the early 1800's, even as the War Dept. tried out various designs of private manufacturers. When the Civil War did break out, the War Dept bought breechloaders- but mostly for the cavalry: there was never a hope of making enough for the whole army. And after the War, the War Dept settled on its own breechloading musket conversion- which , modified and improved, became known as the Trapdoor Springfield. And the Armory would experience the downside of mass production: inflexibility. Once all the special jigs, fixtures and tools were in place to build piles and piles of Trapdoor Springfields, it was hard to make something else. It would resist making more modern magazine smokeless powder weapons until 1890.

Merritt Roe Smith: Harper's Ferry Armory and the New Technology