The pistol I have when fired near livestock creates a wave of panic in the barn which does not settle down for days among the stock. Not to mention it does not always deliver instantaneous results, thus needing more shots fired. The hammer is instant, merciful, and personal.
The girls were not big hunters. The freezer always had plenty of meat from what we raised which didn't go to market or sale at auctions.
Federally-licensed suppressed weapons (silencers) takes care of that. Lots of farmers have used them in the past.
In No Country for Old Men, the evil assassin Chigurh uses a stock-culling "tool" of that type.