Sadly judging by various comments here I don't think most people really grasp what the author is talking about. Not that it is odd. I also failed to grasp this most of my life.
In fact I ridiculed play oriented schooling. It was not until I became a father myself I got to observe for myself how children learn and see what a disaster school was that I finally understood what a lot of these "radicals" where actually talking about.
It is spurred me to discuss this with many other parents. I see so many of the same problems. School really has a nasty habit of killing the joy of learning. In its defense it is an okay substitute when parents have no love of learning themselves and don't encourage creativity and curiosity in their own children.
The key problem I see is that schools obsess about teaching X amount of material. Rather than focus on understanding. A child struggling with learning concepts in math will choose memorization as an efficient strategy because covering all the needed material by actually learning to understand it would take too long time.
This sort of quantity obsession means memorization almost always gets picked over genuine understanding. Actually learning to think critically and clearly is not prioritized. Sure they always claim that is prioritized. But priority implies something else is prioritized less. And that is never the case. Regardless of the flowery words, getting through the whole curriculum is always prioritized above all else.