I have 4 kids, 4,6,8 and 10. I love taking my kids hunting with me.
Here is what has worked for me. I started them all out at 3-4 by letting them "go hunting" with me while letting them carry a specific blue toy pop gun when I went squirrel hunting. The purpose of the blue pop gun was to teach them from day one to have impeccable safety with the muzzle and trigger when walking thru the woods. We would both line up on a squirrel and we would both fire at the same time. To them it was a real rifle and if we were not hunting we kept it in the safe with my guns to further the safety point.
They progressed from the toy to lots of reactive target shooting (to keep it fun!) with the 22 and 17HMR and once they were hitting consistently we went hiring for real for squirrels and rabbits. The first couple times we only took their rifles so that I could really prioritize them and help them pay attention to the details.
Only my two oldest have gone deer hunting so far and they had both just turned 8 the first time. However, they had a ton of ground work and we were both confident in their shooting ability. That being said the first time they ever pulled the trigger on a center fire was on their first deer. My son used a 30-30 and my daughter used my 6.5 CM with a brake. They knew how a 22 rifle worked and they applied the same fundamentals to the hunting rifle. Zero issues with recoil as any recoil was instantly absorbed in the excitement of the moment.
Due to the hard butt plate on my 30/30 I don't think I'll use the 30-30 again for a first rifle for a youngster. A .243, or a 6.5CM with a break and a nice limb saver are much better choices.... but dad or mom may have to help carry it if it's a long hike in.
Best of luck and great job at intentionally creating memories with your kids!!