Rimfire came first, and .22 Short has the record for the oldest cartridge still made, introduced by Smith and Wesson in 1857.

Rimfire ammo had a great advantage over the later Boxer and Berdan primed centerfire cartridges: it was cheaper to make. But it could not be reloaded - a major consideration for the 19th c. shooters, who were used to doing that. It was weaker: the case was thin at the base, and that meant not only could there be a gas leak when the rim was crushed, but extracting a spent cartridge case from a fouled or dirty chamber could sometimes result in the base of the cartridge case being torn off. When smokeless powder was introduced circa 1890, larger caliber cartridge cases also could not easily handle the increased pressures if they were rimfire, or similar "balloon-head" construction. Small ones, like 22 LR, didn't have a problem with extraction or gas leakage: at a small scale, rimfire works fine .

Through the end of the 19th c., there were a fair number of rimfire cartridges still in use, all blackpowder ( except for .22) . Old catalogs around 1900 had around 42, but those were quickly winnowed down to 17 by 1930. Some of those, like 25 Stevens, actually apparently worked pretty well. But there were few largeer calibers: 41 Swiss was there because huge numbers of Vetterli rifles had been imported around 1920, and their owners of course were willing to buy the ammo. But there were very few owners of Spencer repeaters needing 56 Spencer, of henry rifles needing 44 Henry, and as more and more of the old rimfire rifles and pistols were replaced by more modern ones, the ammunition makers loaded less and less. Now the only rimfire calibers are .22. It is hard to imagine making a version that's centerfire without great increase in cost, and a good bit of the case that's now space for holding powder would also be lost in making a solid base to hold the primer, so it's not likely to happen.