But he did a followup study involving more countries to see if most or all high-performing countries used the Japanese approach. It turned out that they did not. Some were more like the US than the US.
Here (http://timssvideo.com/sites/default/files/Closing%20the%20Te...) is one place where Stigler reports his updated findings and attempts to debunk your (and my) initial conclusion---a conclusion that seemed strongly justified by his 1999 book---that, as you posted above, "The more important difference between mathematics education in Japan...and the US is...the encouragement to pupils in Japan to try to figure things out for themselves."
He points out that another high-performing country in his second study, Hong Kong, was more US-like and less Japan-like on this spectrum than the US itself. On the dimension you're calling "more important", the low-performing US is between the high-performing Hong Kong and the high-performing Japan.
His conclusion was that the main factor was not having kids figure things out for themselves but having teachers carefully teach kids the relationships among things. It didn't matter if the US kids spent time practicing procedures. The Chinese kids spent MORE time practicing procedures and did better, but then the Chinese teachers spent time directly pointing out important relationships, which the US teachers didn't do much of. The Japanese kids had to spend a lot of time figuring things out for themselves, but then the teachers would gather them together and carefully lead them to see relationships that they hadn't seen when working by themselves. The US teachers would tell kids to figure things out for themselves and basically leave their learning to whatever they managed to figure out.
Given equal IQ, time on task, etc., it's the effectiveness with which mathematical relationships are made clear to the students (part of which requires significant procedural drill, which Japanese kids do after school) that matters most. A lot of time is wasted in the US doing procedural drill with no conceptual understanding, with even more wasted on constructivist "discovery" methods whereby kids are supposed to somehow teach themselves and each other the mathematical relationships, and all of this led by teachers who aren't required by their union to even know anything about mathematical relationships much less teach them.