Right now hunting is a very narrow gameplay system. Bows do very little damage to animals even with the Hunter. So more often than not hunting consists of chasing down animals on foot with spears and stabbing them to death. The only additional mechanic is that you harvest the game then and there with a knife. The Butchery mod adds in some good ideas but i feel some of the mechanics are gamey(no pun intended). Here is what i propose to add progression to hunting and make it more immersive
Add critical hit spots for animals like the head and heart. Make it so you get rewarded for setting up a good shot rather than running them down with a spear(This also may require making bows more accurate)
Make it so animals bleed slowly if a damage threshold is reached. This mechanically accomplishes two things. One, it makes it so bows can kill something like a deer in "one hit" if the shot and weapon is good enough without making bows mass murder machines. Two, it makes it so that larger animals become easier to hunt as you unlock better weaponry thereby tying it more directly to the progression system. It also ties in with change 1. by making it so hitting a critical hit spot with a weaker weapon allows you to reach the damage threshold when you otherwise wouldn't.
Add in hooks and the butchery table like from the Butchery mod(with a small tweak). The most immersive addition of the Butchery mod is that it makes hunting more of a complete process while simultaneously more rewarding. You *can* just use your knife but you will get less rewards than if it's butchered properly.
3a. Instead of having better hooks and tables give you more meat or hide which is very gamey. Make it so that you require stronger hooks for larger size game. A rabbit or a fox can be skinned with a primitive hook, while a bear will require an iron hook. This further ties the progression system of the game into Hunting.
3b. The penalties for using just a knife instead of a butchery table and hook should be smaller for smaller game and larger for larger game. This makes it so you don't feel totally obligated to setup butchery equipment immediately, while acting as a punishment later on for not doing things properly and trying to bypass the progression system.
To balance out these mechanics, make it so that hunting takes much longer while simultaneously making it more rewarding. Hunting a deer should take an entire in game day and a couple more to process, but should provide enough meat to last several days.
Overall this would make hunting a more involved, fun, immersive, and realistic process. It would require about the same time and difficulty to hunt in the earlier game but require much more time and effort in the later game, with higher tier weapons and tools required to make it worth while. It would also still allow flexibility by allowing you to get some meat on the spot from larger game.
(Bonus?) Add in stamina to animals so you can hunt the way ancient humans did through what's known as Persistence Hunting.