"Stunning" refers to the methods of attempting to render an animal or bird unconscious prior to slaughter. The main methods used in the general slaughtering industry for cattle and sheep are:
• captive bolt gun: a steel bolt is shot into the skull at the front of the animal’s brain.
• electric shock: electrodes are clamped to the animal's head/heart and the animal is electrocuted.
These methods are contrary to Jewish law, because an animal intended for food must be healthy and uninjured at the time of shechita. The above stunning methods injure the animal, making it treifa (non-kosher and thus prohibited). If the stunning kills the animal it makes it neveila (an animal which has not been shechted) and is forbidden as food for Jews.
With these methods, during the delay between the stun and sticking or cutting, the animal can regain consciousness, as has been reported by animal welfare groups. The stun effected by shechita is irreversible and there is no delay. Shechita therefore, is humane and efficient.
Apart from the halachic prohibition against the above methods of stunning, there is no conclusive evidence that these methods render an animal insensible to pain. There is evidence that they are only paralyzed, and thus prevented from displaying their pain. Furthermore when the captive bolt method fails, as it does in a significant percentage of cases, it causes considerable additional suffering and distress to the animal. In such cases, the conscious animal is in acute pain as the captive bolt gun is reloaded and reapplied, or the electrical tongs reapplied to re-stun the animal. Shechita avoids these problems of ineffective stunning, since there is no mechanical or electrical appliance to be misapplied or go wrong. Shechita produces an effective and irreversible stun as well as being a humane and efficient slaughter method.

You stated, If the stunning kills the animal, it makes it neveila (an animal which has not been shechted) and is forbidden as food for Jews'
Did Hashem disagree with this statement with His laws of shechita found in Leviticus 17:13 and Devarim 12:21, “And if any Israelite or any stranger who resides among them hunts down an animal or a bird that may be eaten, that person shall pour out its blood and cover it with earth. If the place where יהוה has chosen to establish the divine name is too far from you, you may slaughter any of the cattle or sheep that יהוה gives you, as I have instructed you, and you may eat to your heart’s content in your settlements.”
If Hashem approved of hunting allowable animals, like Jacob asked Esau to use quiver and arrow, are we misunderstanding G-d's words, as He instructed you?
The Hebrew words in Leviticus 17:13 that you translated as "hunt" is actually "yatzud," for which the correct translation is "trap." Trapping is the only way to catch a wild animal or bird before slaughtering it, as opposed to a domesticated animal which requires no trapping.
Thanks for your reply.
You infer "hunt" is actually "yatzud" for which the correct translation is "trap. This is confusing as Jacob tells Esau in Genesis 27:33 says,
וְעַתָּה֙ שָׂא־נָ֣א כֵלֶ֔יךָ תֶּלְיְךָ֖ וְקַשְׁתֶּ֑ךָ וְצֵא֙ הַשָּׂדֶ֔ה וְצ֥וּדָה לִּ֖י (צידה) [צָֽיִד]׃
Take your gear, your quiver and bow, and go out into the open and hunt me some game.
Did Jacob give his son permission to hunt (yatzud) with bow and arrow instead of trapping and entrapping allowable game?
What are we to believe?