If it was mine, I'd shoot it, but only with black powder loads. Black powder factory cartridges are pretty readily available, but they are expensive. While I consider the .41 Colt a historically significant cartridge, it does have a rather bizarre configuration; i.e., an oversize bore as standard requiring the use of hollow-based bullets in "inside lubricated" cartridges. For such, I think black powder would probably deliver better accuracy than any smokeless propellant.
The generally accepted cutoff for Colt SAAs supposed to be safe with smokeless powder is serial number 192,000, which means production about during or after 1900 (see Kopec, Venturino, others). Your revolver is pretty close to that cutoff. And I do remember reading--in Mr. Kopec's book, I believe--that there was a significant SAA metallurgical upgrade in 1905. A pretty good case for confining the revolver's use to black powder. Research in old ammunition catalogs from the pre-1920 period will reveal that up until about that year, about as many black powder factory cartridges were still being sold as were smokeless factory cartridges. I wouldn't be at all surprised if such sources revealed that smokeless powder loads weren't yet available for the .41 Colt in 1901.
The mantra of "Have the gun checked by a competent gunsmith familiar with old firearms." offers theoretically very sound advice. Now have fun trying to find such a person. Most gunsmiths are competent enough to replace parts in AR-15 rifles and busted automatic shotguns, and that's about it. It has been my experience that such people will tell you that you must be clinically insane for wanting to shoot any firearm made before 1940. Few understand black powder cartridge weapons at all. You're more than likely pretty much on your own if you decide to shoot the revolver. If the major components are sound--no cracks in the frame, cylinder, or barrel; hammer and trigger interface works properly; lockup is still sound, etc.--it's almost certainly safe to fire with black powder loads.
The risk of parts breakage? If you fire it enough, I suppose you might very well run the risk of eventually breaking the trigger and bolt spring (which usually doesn't require any special fitting at all) or, less likely, the hand spring, which does require the service of a competent gunsmith but isn't a devaluing catastrophe. Other than that, you have to inflict some fairly serious abuse on an SAA to hurt it.
At the end of the day, I know you're not going to use your old SAA as any sort of primary weapon or subject it to any abuse, so go ahead and astound the Glock crowd at the range with some fire and smoke.
Don Kenna