Having this week spoken to a slaughterman of many years experience, he told me that the method for dispatching of horses is different than that of cattle. For instance, NEVER stand in front of a horse when administering the shot, they have a nasty habit of sometimes leaping forward onto the shooter! Whereas, cows go down like a ton of bricks. Also, he confirmed that the pistol is silenced as to not panic other horses in, for instance, stables, that may cause a broken leg or worse.
In conclusion, i think this weapon is for use by a person who works with or cares for horses, and as such is a specialist piece for expert use only.
Saying that all soldiers would be familiar with horses, and so could be expected to carry/use this weapon is obviously incorrect. As an analogy, most of us drive vehicles today, but how many can repair a simple breakdown? Not as many as you may think!
Your thoughts are appreciated, Ned.