What about kids who, left with no structure, will simply do nothing at all? I was one of them and I would certainly benefit from a structured program. Hearing all that's said about the current children's screen time and seeing all the young people walking around with some degree of lower crossed syndrome makes me think it has only become more common since I've grown up.
I think you have the dichotomy wrong. It's not a choice between just three movements and a multitude, but between three (of however many there will be in a chosen training program) or zero (which is what will likely happen if the parents or other adults don't provide some structure). Whatever the merits of S&S, I'm absolutely certain it's superior to sedentary living. And if you consider it insufficient, then the answer should be a program that includes more movement patterns and more self-regulated programming, not passively avoiding the issue in a hope it will sort itself out somehow.
Better is relative, and in these times, it doesn't mean much.
Bottom line: you just can't expect Kids These Days to figure out how to move on their own. I'm barely over 30 and all the older guys' (with all due respect) talk of "play" and all the other things children supposedly do all day that obviates the need of a structured training program makes me feel like I've recently arrived from Mars. Even in my own childhood TV and computers were already establishing themselves as the default free time activity. And right now, I imagine, the internet, video games and Netflix are just more interesting and more comfortable than anything you could possibly do outside.
My own upbringing generally predisposes me to dismissing attitudes such as "you shouldn't push things on kids". Providing a structure to life is what parents are FOR, in my opinion, and having a father who lifts is a tremendous asset to
@Nate 's kids that should be fully utilized instead of shied away from.
That said, I would start with bodyweight training, NW style, with possible inclusion of pullups and maybe kettlebell deadlifts. S&S lifts may be dangerous (the kettlebell doesn't just look like a cannonball, it can quickly transform into one if handled by someone careless), 4 kg jumps too big, and the prescribed progression too quick.