Are there any private all-boys[1] schools that embrace this side of childhood? Look at tiger cubs and what not: they play, they bounce around, they fight, they figure out the world around them through investigation and exploration.
I spent so much of my time outside as a kid, but I also spent a lot of time inside on the computer. Both were amazing worlds where your imagination could run wild; especially in the early days of computers and the internet... being a part of a "wild west" frontier at the age of 10 was a special privilege.
My point is that learning and running around in the mud are both very valuable and can be balanced to create well-rounded people who have confidence with technology and the outdoors.
Has any institution figured out how to balance these two in a formal curriculum? Spend half the day exploring the woods and classifying wildflowers, the other half back inside working with math formulae and reading lessons.
Some might say there's not enough time in the day to spend half of it outside, but I disagree. The kids will be more efficient in their inside learning time if they've spent half the day learning outside.
[1] I'm not saying that girls wouldn't appreciate the same type of learning. I suspect that in an environment that harvests the physical energy (horseplay, outdoors activities) of boys to improve learning, even in a positive and productive manner, might alienate some young girls who might feel uncomfortable, so perhaps it's better to separate the genders in this sort of environment?