I have walked in your shoes, bringing kids up to shoot and balancing their experience and recoil sensitivity with the guns I have available for them.
Let me put out there that every kid is different. You CANNOT expect that because you could or could not handle recoil at their age that your kids will do the same. I'm not saying you need to coddle them, but don't expect a cookie cutter experience. Each can learn differently and experience recoil differently.
In anticipation of my oldest one joining me in the deer and hog woods, I bought her a Weatherby Vanguard Youth in .243, using the same logic of many of the posters have put here. She shot her first hog and deer with that rifle.
Because of that, it came as a huge shock to me one day when I went to pack up the range gear, ammo, and rifles for what should have been some fun father-daughter shooting time that she said she didn't want to go and didn't really like shooting...
Dejectedly, I sat down with her and had a long talk about what made it so unenjoyable. It was the rifle. To her, the recoil was uncomfortable. To me, it was a joy to shoot!
Realizing I needed to break my paradigm of a "kids deer/hog rifle", I came to TFL and posted my experience to gain some perspective and seek help in solving the dilemma. In the end, an the AR platform made perfect sense! An adjustable stock allows it to grow with the kid. Keeping it in an AR-15 vs the AR-10 sized rifle (surely Tim77W will be along shortly
) kept the weight down. The action with that buffer and spring was the logical choice for recoil reduction.
Another huge plus was the modularity. Practice is important. Having the ability to shoot cheap plinking ammo is a bonus. However, I am not a fan of .223/5.56 for deer, and with my job, moving is always on the horizon and not all states allow you to use it. So, the ability to swap uppers became a real bonus. I chose 6.8 SPC, but there are others that are capable of providing more knockdown than .223/5.56.
Fast forward to today and my 19 year old LOVES her AR. She's taken 5 deer with it, including a 9 point that is going on the wall. She enjoys going to the range to shoot with dad as well as show up the boys.
Another bonus to the parents of today are the plethora of managed or reduced recoil offerings in so many calibers. Those did not exist when I went through this.
So as you proceed down this path, consider the needs of your 10 year old. You know him more than any of us.