The 1855 Springfield was basically the US Ordnance Dept copying the idea of a .58 rifle-musket from the British Enfield concept. The original Minie was .5775 because during peacetime, the 1855 Springfield was a very finely made rifle with close bore tolerances.
When the war started, all of this went out the window as 12+ contractors were making 1861 Pattern rifles, the US was buying Enfields, the CS tried to standardize on the P53 while also making a copy of the 1861 Springfield.....very confusing and not conducive to marksmanship training. There was never a "standard" as there were so many calibers and types of rifles being used it was hard to supply ammo for every different type let alone train with them. The .58 Minie/ Burton was constantly being altered and lube formulas changed, by the US Ordnance Dept and the Confederates tried to standardize on the Pritchett cartridge, but there were also other types of bullets and cartridges used.
By the time ANY military fully figured out the rifle-musket, technology had advanced and by the late 1860's every major power had gone to Breech Loading rifles. The British , hands down, get the #1 spot for mastery of the Rifle-Musket in actual combat. America (and the CS ) never were able to fully use them to anywhere near their potential.
4-6" at 100 yards was considered acceptable for a .58 rifle-musket. This would still allow hits on a man at 300 , most of the time and this was good enough for wartime use. And 4-6" was from a sandbag rest by an experienced shooter. The vast majority of people, then and today, can't pick up a .58 rifle-musket and shoot into 6" at 100 yards from a standing position, or any position, even if someone else loads for them. "Accuracy" is over emphasized in my opinion. Acceptable Combat Accuracy, Range and Reliabilty are the real concerns because absolute Accuracy can only be effectively used by very experienced and trained shooters. A battle hardened Regiment like the Irish Brigade armed with .69 1842 Springfield muskets would absolutely rout a regiment of the same size, of green, inexperienced soldiers with P53 Enfields because the men with the P53's and no training wouldn't be hitting, they would be panic firing while the salty, hardened men with .69 Smoothbores would be at 100 yards, hitting the other men with "accurate enough" fire. Accuracy is useless if you can't properly use it.
By comparison, a military M16A2 that shoots 3" at 100 meters is considered within spec and means the rifle is mechanically capable of hitting a man-sized silhouette to 300 meters. And even that is beyond the capability of the majority of shooters using this type of rifle. "Mil Spec" doesn't equate to accuracy, it equated to "within spec" of a mass produced weapon that performs well enough for the shooting that will be expected of it. I myself would be dissapointed if I saw 3" from an AR15 I owned.
I fired maybe a 10" group at 100 with my ArmiSport CS Richmond a few days ago, with some random .575 Minies I found that I sized a few years ago. Obviously the rifle would probably like a larger bullet but in 1863 that rifle would have been good enough for Govt Work....it fired and functioned with .575 Minies and put them where they needed to go. The chase for MOA accuracy often leads down a road of frustration, and sometimes you just get what you get with that weapon and ammo.