All cats play with prey. Thus, it is evolved behaviour. It has some benefit, otherwise cats would just use the killing bite straight away.
> Paws are able to be bitten [...] there is no benefit to prolonging the exposure to danger by keeping the prey alive.
Cats have 4 paws. It's possible for a 3 legged cat to survive. Hard, but possible. The cat prolongs the exposure to danger of redundant limbs in order to protect the jaw.
> A quick bite to the neck is the best chance of killing. Letting the prey struggle and escape to be re-caught is lowering the chances of getting food, not raising it. You are claiming the opposite of reality.
A bite to the neck is the method of killing. Small animals are quick, thus the cat plays with the prey to weaken the animal so that when the cat applies the killing bite the small animal is less likely to escape.
> They only do so when they are teaching young to hunt. They do not play with their prey when they are engaged in the process of "acquire food to prevent death".
This is incorrect.
EDIT: You are incorrect about only one research suggesting that play with prey is a defensive part of hunting behaviour.
Trivial www searching find many different researchers suggesting this.
> In this article, we show that feline predation involves a continuous gradient of activation between defense and attack and that predatory "play" results from an interaction of the two. [...] In such shifts, no sharp demarcation between play and predation was evident. [...] These results suggest that play with prey is a misnomer for predatory behavior that fails to escalate along the gradient between defense and attack. Movement notation analysis revealed that playful movements are adaptive in that they protect the cat from injury.