Eltigra;
I too love the American single-action revolver designs, and I own and shoot modern 'clones' of both the 1873 Colt and 1875 Remington. One of the main reasons for the strong appeal of Cowboy Action Shooting (including the many participants living outside the USA) is that fascination with the American Old West (both fact and fiction) largely attributable to Western movies and television shows. Indeed, I often liken the sport to "grownups dressing up and playing 'Cowboys and Indians' with real guns" .... though, of course, we don't shoot
at one another. SASS (the Single Action Shooting Society) recognizes the strong movie influence by allowing shooters'
personas to be either "Historical" or "Hollywood" .... or, as it is sometimes put, you can depict a "real cowboy" or a "reel cowboy".

(Mind you, another nice thing is that you don't actually have to be a "cowboy" - after all, there were lots of other walks of life in the Old West, and eventually you'll likely see someone portraying every one of them! Rather more "all-encompassing" terms for the sport are "Western Action Shooting" or even "Frontier Action Shooting" ....)
However, SASS permits the use of single-action revolvers only .... and it must be remembered that the S&W Model No. 3 (and the Schofield variant) were also single-action, which is why modern 'clones' are now available. As already noted, back in the latter half of the 19th Century, the British had developed a decided preference for
double-action revolvers. The much smaller National Congress of Old West Shootists (NCOWS), to which I belong, permits the use of any handgun design which was in common use in or before 1899. In shoots held under NCOWS rules, there are accordingly lots of 'period correct' double-action revolvers in use.
I normally use a good, top-break, double-action Webley revolver in events sponsored by the Grand Army of the Frontier (GAF) - which are shot under NCOWS rules. GAF is an organization which beautifully combines my two primary passions: 19th Century military history/re-enactment and frontier action shooting. Although "non-military" shooters are encouraged to participate in the "Scout" class, GAF is geared toward competing in a historically correct Victorian-era military
persona using firearms appropriate to that impression. Rather than the "four gun" (two pistols, pistol-caliber lever-action carbine and shotgun) battery of firearms normally used in SASS and NCOWS, participants in GAF events are limited to one pistol and a rifle, which is certainly more in keeping with military usage. There is even provision for shooting in categories where only a single firearm is used, such as "Staff Officer" (handgun only) and "Infantry" (mil-spec rifle only).
For chuckles, here are some pics of me participating in GAF events -
As an 1890's Royal Highland Regiment (Black Watch) officer -
In my 1885 North-West Rebellion Queen's Own Rifles of Canada
persona -
This was a delightful "amphibious assault" scenario, in which each participant fired from the moving boat at a series of gongs on the shore. The fellow with me (dressed as a Spanish-American War U.S. Marine) joked that, despite having been in the Marine Corps for real for 20+ years, this was the first time he had ever had to shoot from a boat!