Cats generally don't have the need to chew on random objects like that. He's not a dog. Cats use their teeth and mouth pretty differently. In fact cats do very little chewing. It's usually shredding and swallowing. Your example of the tug-of-war when he catches his prey is actually a very "catty" behavior. They want to take their prey off to some place where they have some privacy (and not be disturbed by another predator) and enjoy the meal. That is a strong cat instinct, so yea, he doesn't want to easily give up his newly caught prey. And unlike dogs, my take is that I don't believe they actually want a tug-of-war, what they want is to play out that "carry off the prey" thing. After doing that, they gladly give up the prey to re-start the game.
For a short time, in kittenhood, my bengals wanted to chew on wires and cords. I bought these cord covers with grooved ridges, and then sprayed them with bitter yuck. That cured them almost instantly, and they didn't go off and substitute some other object for the ones they could no longer chew. Whew. What you've got to hope is that your cat doesn't have, or doesn't develop, full-blown pica. That would be bad.
So I dunno. I guess I would at least try to dissude him from chewing on shoes and other inappropriate things like that. You can use the bitter yuck spray trick for that.
Some people and companies do make things for cats to chew, like this:
https://smile.amazon.com/Dental-Health- ... ntal+chewsI even bought a few of those products years ago, thinking that it might help with my bengals' dental issues. No interest in them (which in hindsight might be for the best).