I think every kid is different and every parent's approach to introducing and involving their kids in the outdoors is different.
My oldest is 6. I had him riding on my shoulders checking trail cameras with me in the fall, scouting geese after work during waterfowl season and listening for turkeys at 2 years old. In our house, hunting and the processes/work associated with chasing critters is a 52 week a year lifestyle and I'm fortunate to do "hunting stuff" several days a week during the offseason in conjunction with hunting multiple times a week when seasons are open.
At 4 years old he began deer hunting with me regularly and he watched me shoot 5 does that season.
Because he was so involved in the processes associated with hunting, he naturally transitioned quickly from a spectator/observer to a participant.
He killed his first two turkeys at 5 years old from ground blinds on tripods and a 3rd last year from ground blind on a tripod shortly after his 6th birthday. After his 3rd turkey I told him we were no longer hunting from blinds, or using tripods and we weren't hunting with decoys anymore. He hunted several additional times with birds coming in range but he struggled to see them while sitting in my lap and without decoys to put the bird in perfect position. Because of his success, I felt he needed to experience some failure and learn the skills necessary to become a real turkey hunter.
He killed his 1st deer this fall at 6 years old, as well.
My recommendation is to make sure you have the right gear and the right gun to set your kid up for success. Cold weather means blankets. Snacks and drinks are imperative. When it comes to turkeys, roosting the gobbler the night before and setting up close by ensures an action packed experience that starts immediately the next morning. Additionally, the most important aspect of kids being successful is weapons manipulation from hunting positions. My son spent a ton of time working off his tripod in my trophy room moving his gun around to various mounts and squeezing the trigger.
I've taken a similar approach with my 2 year old and he seems to be as addicted if not more than my oldest was at 2. We'll see how it all works out but I have accumulated an incredible number of memories with my son with an integrated approach.
My oldest is still 6; we leave Friday morning to head to Alabama for his first out of state hunt. I'm beyond excited to have my sidekick in the truck with me to kick off my season.
Best of luck to you.