Honkyoku as number 1
Via the Shakuhachi Forum, KITAMORI Shinsuke's MA thesis: Shakuhachi Culture Taking Root in the U.S.A: The Construction of "Japanese" Authenticity and the Lifeblood of American Players [PDF]. tl;dr summary: US shakuhachi players dig the historical connection to Buddhism and premodern Japanese tradition, Japanese players not so much.
[T]he globalization of shakuhachi has occurred in an area that I term the "myth-scape," which is composed of imagined Japanese space Americans have perceived in Zen, bonsai, samurai, ukiyoe, haiku, Noh, etc. In other words, "myth-scape" means "authentic space." Authenticity is mostly different from the actual real situation, but creating authenticity has a significant meaning for people who are involved with different cultures.
I've been thinking about these things for a long time too. The most popular organized shakuhachi tradition (ryū 流) in Japan, particularly western Japan, is the Tozan ryū. This was founded in the late 19th century by NAKAO Tozan 中尾都山 as an attempt to bring what he saw as the modernity and rigor of European art music to the shakuhachi world. Accordingly, the Tozan ryū does not transmit the pre-Meiji komusō pieces most strongly associated with Buddhism, generally known today as (koten) honkyoku ("[classical] core repertore"). Instead, they have new compositions by Tozan and his associates (which, interestingly, are referred to within the Tozan ryū as "honkyoku"), only a couple of which borrow titles or themes from the classical honkyoku tradition. Another huge difference is that most Tozan-ryū honkyoku are for two or more players, while almost all classical honkyoku are played solo.
Contrasting with the Tozan-ryū you have various modernized komusō traditions, the most well-known of which is probably the Kinko ryū. Their honkyoku tradition began as a collection gathered from Buddhist temples around the nation by 18th-century monk KUROSAWA Kinko 黒沢琴古, and even today Kinko-ryū players resist attempts to force the round peg of their music into the square hole of the European art-music tradition.
There is resistance and there is resistance, of course; some players prefer instruments that match European scales and concert pitches so that they can jam with non-shakuhachi players more easily, some favor older instrument designs with less compatible tunings, and others follow aesthetic ideals leading them to reject even late-Edo developments in shakuhachi design like smoothed-out bores in favor of what they understand to be more authentically komusō (or fuke) designs. All, however, share the understanding that they are part of an unbroken tradition stretching back through the centuries to a time when shakuhachi was part of a religious practice.
Now, min'yō (folk song) players are also a significant presence in Japan, at least as important as the organized ryū. Min'yō are an important part of regional identity in Japan, and arguably the most accessible and therefore marketable of any traditional Japanese musical form. But min'yō have no universal component: if you're from Akita, then Hideko bushi might evoke all kinds of feelings in you, but if you're from Ohio, it's just another pentatonic folk song.
So while a Japanese person might take up shakuhachi because they want to participate in their local (or national) min'yō scene, i.e. the instrument is a means to an end, non-Japanese players who decide to take the instrument up are likely to have been attracted by the instrument specifically, and therefore to some form of honkyoku, which are after all the pieces in which the shakuhachi is most striking and distinctive. And this is self-perpetuating: if no-one outside Japan bothers to learn min'yō, and you are outside Japan too, then even if you wanted to learn it, who would teach you?
So that leaves Tozan vs Kinkoid — but it's not a level playing field here either. There are fewer Tozan teachers outside Japan, which means that there are also fewer students and fewer people to play those multi-flute compositions with. Kinko teachers are easier to find, the Kinko repertoire is more solo-friendly, and most of the resources available in libraries and online are Kinkocentric. And, Japan-specific mythologizing aside, the backlash against cultural imperialism in all its forms has put ideas about "authenticity" and "purity" in the spotlight in a way that undoubtedly favors Kinko (200 years of explicit tradition + misty centuries of wandering monks) over Tozan (invented 100 years ago by a dude seeking to make the music more Western).
Further: if you join the Kinko ryū, your teacher is likely to be seriously dedicated to the shakuhachi, impressing on you the gravity of what you are doing: learning tunes passed on orally for centuries, another link in a very long chain that leads back to a religious community containing many ex-samurai. And, arguably, you can't learn to play Kinko-ryū music unless you internalize certain aesthetic ideals that are specifically Japanese. It's not just about getting the right pitches.
Thus, Kitamori's "myth-scape" — the zen, the samurai, the nature-centrism, the wabi-sabi — is of necessity transmitted along with the music, because the main tradition transmitting the music has made it part of the tradition. This is equally the case in Japan. The differences are that in Japan teaching of cultural issues doesn't have to be so explicit, and (more importantly) the "shakuhachi as culture" diehards are such a minority compared to the vast numbers of "shakuhachi as instrument" (Tozan) and "shakuhachi as ensemble voice" (min'yō) players, who are much rarer outside Japan for the reasons described above.
It may well be the case that the pre-existing "myth-scape," particularly interest in zen, was part of the reason that the Kinko ryū spread faster and wider outside Japan than the Tozan ryū. But at some point along the line, the extra-Japanese shakuhachi community entered a feedback cycle, self-sustaining and self-"correcting." It's surprising, actually, how many shakuhachi players report having no interest in Japan before being struck by the sound of the instrument alone.
Leonardo Boiko:
Hmm… I have issues with “no-one outside Japan bothers to learn min'yō”. It might be true of the U.S., perhaps, but it’s not of Brazil. I mean, I don’t have any real data, but I find it far easier to find a min’yō group (I even used to frequent one) than to find a honkyōku teacher. So far I’ve only heard of (Tozan) honkyōku in São Paulo, but basically every city with a sizeable Japanese presence will have its own min’yō group, complete with a stock of instruments and the occasional competition. Finding min’yō CDs in São Paulo street vendors or specialty Japanese stores elsewhere is way easier than honkyoku CDs. To this day I’ve only seen live honkyoku in special presentations, but any immigrant festival is sure to feature min’yō. And min’yō, together with taiko, is considered to be a gateway into Japanese music, far more accessible to the outsider than honkyoku.
Put in another words, I’m willing to bet in Brazil there are far more shakuhachi players who came from Japanese culture to min’yō to shakuhachi than musicians who came from shakuhachi to honkyoku to Japanese culture.
In my experience, the typical min’yō group won’t teach you anything about shakuhachi notation or advanced honkyoku techniques. Some elderly issei will just teach you to blow notes, and then you’ll be thrown in the ensemble until you learn the groups’ favorite songs by osmosis. Shakuhachi sizes are chosen to match the current singer’s vocal range. I didn’t try to learn singing or shamisen, but from what I could see it’s basically the same teaching method.