For what it is worth, I have been going thru this same decisions for yrs, with my own kids and now the G kids.
The best solution I have found is a TC/Encore rifle, comes in numerous calibers/gauges, both rimfire/center fire, can be made into any caliber that is safe to use in which ever action type, very easy to adapt to each shooter, single shot, very safe, teaches the shooter to be a hunter, make each shot count, etc.
And the big thing is, with the kids and the way they grow like weeds, every couple of months you have to add more to the butt stock to keep them in proper form etc. I have had to add spacers to the G-son TC 2 times in less than 2 yrs of use, and he is only 9 and still growing.
Get a adjustable length stock that is produced by Choate Company, for either the TC/Encore, looks like a Black rifle adjustable type, length of draw fits from 9"" to 16" IIRR, as they grow or add heavy clothing etc, adjust to fit, etc. Then when they stop growing, you can add the standard wood stock if wanted.
I have just put one one the G-son's, 20ga Rem 870, he won at the sheep dinner, added a good recoil pad and he was braking clay birds, he first time out.
Since adding this stock to his TC, I have added them to my TC/Encore rifles, I like the way they change length of pull to fit any shooters etc. Saves a lot of work and headaches etc, IMHO.
Tia,
Don