1863 US Springfield rifle

[ssgross]

A simple mistake in loading can be catastrophic

I think this is exactly the point - but true of shooting any firearm. Where is that video of the guy's brand new springfield M1A that blew up in his face first time at the range? The firearm has no idea how old it is. Age only increases the chances of wear to a point outside serviceability. Carefully inspect all components are serviceable, and load with caution, starting at minimum safe levels. Shooting it has all the same concerns as any other firearm - make sure there are no cracks in the stock, bulges in the barrel, excessive pitting, active rust or obstructions in the bore, etc. Use proper projectiles - soft cast (20:1 or 30:1).

To have that old warhorses stock splinter at the range would be a crime.

Most certainly, but not necessarily just because of age. These old rifles and muskets were designed with stocks bedded their entire length, and even when brand new they were they brittle as a twig by design - until the whole assembly is properly mounted and clamped together. The full-length-bedded designs persisted all the way through the Krags and into the early 1903's. By the time of 1903 production, Springfield had learned a thing or two experimentally about bedding and how it affects accuracy. Lots to read about early practical understanding of barrel harmonics with the 1903 and the resulting bedding procedures - Hatcher's notebook has the details.
My point is, when considering shooting an old rifle, be sure to check for any loose fitting from age - from shrinkage of the wood over the years or possibly from fibers soaked with oils (lubricant, not finishing oils) that have deteriorated as a result. The forces from firing a loose gun will exploit the weakness of the design and ruin your stock. You can remedy to an extent (before firing!) with careful shimming to make all tight again as it should be. Other than these issues, or maybe the other oddities of fire or water exposure/damage, the wood doesn't know how old it is either.