As for why armies didn't replace their muzzle-loading guns with breech-loading ones immediately, it's a question often asked about the US Civil War, and I answered it here. Basically, it was because they needed a lot of guns to equip an army, and so it took a very long time - and a lot of money- to make new ones.

Making muzzle-loading rifles is simpler than making breech-loading ones. It requires skill and knowledge to make the lock, barrel , and furniture of a muzzle-loader, but as can still be seen in the Colonial Williamsburg Gunsmith Shop, it can all be done by a few craftspeople working in a rather small shop with hand tools, a forge and hand-powered boring machine and rifling bench. Breech-loaders were made in the pre-Industrial world, like the Pauly, but they took an enormous amount of careful hand fitting and so were quite expensive- and so they tended to be rare. This hand-fitting was why their manufacture benefitted from modern machining methods, that could do repeated precision operations. It's not surprising that the first US military breech-loading guns, the Hall rifles, were also the first to be made in this way, in a workshop at Harper's Ferry Armory. The need for milling machines, screw-cutting lathes and other equipment transformed gun manufacturing from what had been a craft in the 18th c., with many gunsmith shops scattered around the US, to an industry dominated by big makers like Colt, Remington, and Savage. It's something that happened to many crafts, of course.

However, "render obsolete" is a broader question, because rifles are used for other things than warfare. For hunting and target shooting, muzzle-loaders did not become completely obsolete. Some continued to use and make them. The National Muzzle-Loading Rifle Association has now plenty of members, and I think all US states have special seasons for hunting with muzzle-loaders- even if many of the modern muzzle-loaders don't look at all like Davy Crockett's rifle Old Betsy.