I grew up and studied math in Hungary and graduated from Fazekas, the school mentioned in the atricle, and its "special math" high school class. I also spent a year in high school in the US so I have some basis for comparison.

My experience was pretty much exactly as described in the article as well and I am forever grateful for the school and my teachers for giving me a foundation I could build on later in life. I'd like to point out a few things though, others have already touched upon some of these:

First, similar to why it's hard to replicate the Silicon Valley startup model elsewhere (or at least why it takes such a long time), the issue is somewhat similar here. The method of teaching is just one part of the equation. It was a whole "ecosystem" of fantastically knowledgable and respected teachers who could anywhere else be university professors or researchers, publications (such as KoMaL), camps, competitions and other extracurricular activities aimed at elementary and high school students, and a culture of math and science being interesting and fun vs. the stereotypical hard and boring.

The way of teaching feels a little bit more of a consequence of this culture rather than the source. You can pick your own preferred origin story for how that culture emerged: a booming industrial economy in the age of the Austro-Hungarian empire, a multi-ethnic, liberal (in the original sense of the word and given the context) country, the migration and resulting concentration of Jews in Budapest, or a series of exceptional teachers and mentors. The highly visible world-wide successes in the first half of the 20th century then later provided an on-going narrative that benefited the national ethos and hence made plenty of funding available in the second half (conveniently forgetting the austro-hungarian, the economic boom, the liberal or the jewish part of the story). But in any case the culture and the support system is (or at least was) there and that one is very hard to replicate at a broad scale, although certainly easier within specific communities. As someone mentioned, even in Hungary it is not broadly present and is limited to a certain set of top schools.

Also, there is a flip side to this story. Personally, I really liked math before going to high school and completely lost interest by the time I finished. Partly because a lot of kids around me were much better so I felt like a failure, partly because I was more interested in computers and programming and also in finance, all of which was looked down upon. The prestige of winning a programming competition was nowhere near the same as placing well in math or physics. Working as a developer on the side was considered a distraction. I think this was for the better for me personally. A decent number of my classmates got burnt out and had severe depressions due to the pressure. You were almost expected to win a gold medal at the International Olympiad and eventually become a world-wide math celebrity. I can't shake the feeling that a lot of them "peaked" at the end of high school, although perhaps that's partly the result of the rapidly declining university system.

Again, I'm very grateful for what I got but it's more in terms creative thinking and problem solving than specific math skills. In fact, I got to learn other subjects, such as history and literature through the same method which I now realize is very unusual in a country where those subjects are usually heavily biased toward insitilling a national identity as opposed to fostering independent thinking.