First off, I'd just like to preface everything I'm about to say by also saying I'm not a father, and I wouldn't presume to tell a father how to handle something like this, or even to make suggestions...I'm only offering reflection on how my own father handled this exact coming-of-age process. In my opinion, he knocked it out of the park, and while we differ even to this day (I'm in my late 20s) on some aspects of the hunt, that's due to difference in personality, and skill strengths and weaknesses in our respective hunting styles.
That out of the way, when I was at the stage your little hunter is in now, I was a bit older (PA hunting laws at that time basically said if you weren't going to be 12 that year, you didn't pull the trigger, no exceptions). Like him, I'd been shooting a BB gun since I was maybe 4 or 5, and a .22 for nearly as long, and was good with both.
Additionally, I had exposure to the realities of hunting from a young age as well, seeing and touching the dead deer dad would bring home, having a collection of old licenses and empty shell cases and a few other bits and pieces of hunting ephemera.
When I was maybe 10, dad started taking me on squirrel hunts. The mid-October weather was nicer, the woods livelier, and the action (usually) more frequent than the long cold december deer hunts. He told me that if I did okay with squirrel, he'd let me come along for a deer hunt or two. For me, being with dad in the woods was the main goal, so I had a blast going for squirrel, even on one memorable hunt where it was pouring rain from the night before, through dark that day. Didn't matter. I was hunting. Made it through the cold days of deer too. (On this note, though, there is certainly a point worth making for a kid losing interest because he isn't actually hunting. For those long deer hunts, I'd have much rather been camping with dad than hunting, but a day in the woods is a day in the woods...a sensibility that I keep to this day.)
When it was time for me to hunt, dad started me with a .410 pump. I was a skinny, small-framed kid, and a lefty to boot, so he didn't want to start by spending on anything specifically for me, until he knew I was ready. After the 410 came a 20 ga single shot. It kicked a bit more, but wasn't a problem for me. The night before one hunt, I went for the 410 instead of the 20 I'd been using, and when dad asked why, I explained that I'd rather have the follow up shots than the extra BBs. Not long after that, he and my grandpap surprised me with an Ithaca M37 20ga (bottom eject for my wrong-handed self), which I've used ever since.