This is basically the foundation of doing a halal slaughter. The animal must:
- Be alive and in good health prior to slaughter
- Not witness the slaughter of another animal
- Be slaughtered with a very sharp, smooth blade
- Be killed by a cut that severs the windpipe and all arteries in the neck
Halal law also stipulates not eating from an animal that has been gored (contaminated meat from gut contents), already preyed upon by wild animals (any number of zoonotic diseases), or strangled/bashed on the head (high stress/inhumane).
Religious lunacy aside, I personally think this all makes a lot of sense. It's a relatively humane and quick death for the animal. They don't see other animals dying, keeping stress low and meat quality high.
A lot of these old religious laws regarding food (kosher, halal, etc) make sense if you look at them in the context of their era. Maybe eating tapeworm infested pork or vibrio containing shellfish is a bad idea if you don't have modern cooking standards or refrigeration. It's all just Abrahamic ServSafe training.
Edit bonus: Jewish law on slaughter is very very specific on how the cut is to be made across the throat. tl;dr One smooth slice, no tearing, no stabbing, no stopping. Read more
here.