From Nintendoeats on Utreon:
"Modern centerfire and rimfire cartridges seem (to the casual observer) like they would always have been the simplest cartridge types to make. Why was pinfire ever used, and how was it economically manufactured?"

Essentially, pinfire represents a cartridge development that predates the technology to draw brass cartridge cases. Pinfire developed from the first Pauly self-contained cartridges, and in its original iteration is used a non-obdurating case head combined with a paper body. It became popular because it was a convenient fit with the existing firearms architecture - the hammer required to fire a pinfire cartridge could be nearly identical to the hammer from a percussion lock.

As technology around brass improved by the 1850s, solid brass cartridges became possible. At this point the all-metal pinfire cartridge evolved, and became popular driven by Eugène Lefaucheux as a continuation of his father Casimir Lefaucheux's original system. At the same time, many other cartridge systems were also in development. Pinfire got a boost from its existing manufacturing and intellectual property foundations, while other cartridges had to create their own new firearms platforms.

In the US, George Morse (of Baton Rouge, not South Carolina as I say in the video - sorry) invented the predecessor to the modern centerfire cartridge and primer, but his work was interrupted by the US Civil War, and further by Morse's allegiance to the Confederacy and thus his difficulty in refining his invention during the war. Eventually, though, the most efficient system - centerfire - came to dominate the firearms landscape.