In that recent Paul Harrell video about giving guns as gifts, he discusses examples of maturity that are more meaningful than a specific age number. I would add that you don't have to give the gun as a gift to take someone shooting, but the video was a timely discussion related to gifts. It could just as easily been about when a kid or anyone else is ready to be taken shooting.
I don't think that kids younger than teenagers who aren't showing an acute interest in shooting will miss a lot if they're not taken shooting. By the time their well into their teens, they probably need to be taken shooting simply as a matter of cultural literacy, life skills, and to train them properly before they encounter opportunities to make mistakes out of ignorance, like when their buddies bring the guns out.
By waiting until they're closer to 12 instead of 6, they're likely to have guns fit better, able to deal with more recoil allowing them to shoot a greater variety of guns, they'll be able to hold the guns with less aid, and they'll be able to gain meaningful skills in fundamentals like sight alignment, sight picture, trigger control, breath control, stance, etc. And if they actually hunt anything, they'll be able to use guns that get better results, like 20 ga instead of .410.
I would also add that lead safety is more critical to younger children than it is for older ones and adults. Rimfire ammo is the filthiest cheap crap. I can't think of a reason why kids should shoot rimfire ammo. There isn't one. Not one.
I only shoot centerfire with Fiocchi ZP lead-free primers. I've shot over 1500 of them now with not one failure. My guns, case cleaning media, reloading bench, and workshop are practically free of lead styphnate residue.
If I was going to introduce an a child 8 or so to shooting, I would start with a semi-automatic MSR or bolt-action LRP style rifle with adjustable length-of-pull clamped in a tripod with a sight or optic that allows for the necessary eye relief (regular eye relief scopes would need to be mounted forward on the rail. Red dot, peep or open sights also work, but consider whether the child has myopia, astigmatism or corrective lenses). Lots of cartridges can be made for low recoil using lead-free primers and bullets with no exposed lead. Most pistol cartridges are low recoil in a rifle and rifle cartridges with smaller case capacity (from the Hornets up to 6.5CM) can be made low recoil with Trail Boss powder. Obvious choices would be 9mm or .223. But you could also shoot .450 Bushmaster, 6.0mm Creedmoor, 6.5 Grendel, all of which can be loaded very light or up to excellent performance on deer. .357 and .44 Magnum can also work from light loads for beginners up to deer-loads, but you'd probably have to cut the stock on the available guns. I'm attracted to the Ruger M77 .357 bolt-action. In my experience, young kids find lever-action rifles very front-heavy. The weight and balance has shown to be difficult for them.
For handguns, I think revolvers in single-action are among the best for little kids. The .32's and .38's work well. I'd consider longer-barreled S&W Model 60, Model 66, SP101, Ruger Single-Seven. Consider the weight.
