I think this question deserves an answer, but it's not going to be a normal answer. The normal answer is basically that the UC in Japan is very effective at making money in various businesses and at directing that money towards the organization, which has used it to try to influence Japanese politicians, as well as funding the headquarters in South Korea and missions overseas. But how did it get so powerful, and why have politicians been so accepting of it? This is an unresolved argument, and I'd like to explain why that's the case.
There is a gap in the telling of post-1945 Japanese religious history. Religious groups that have opposed the Japanese state, both wartime militarism and the postwar state, have been fairly well-documented both by religious scholars and other historians. Groups that embrace the ruling political party and the state are documented almost vanishingly poorly. This has happened for different reasons in Japanese and Western language academy, and I won't get into that.
Following Abe Shinzo's assassination, a scholar named Tsukada Hotaka became the go-to critical historian. Tsukada has made an academic career out of criticizing religious involvement in Japanese right-wing politics. In his publications, he highlights the number of business fraud and misconduct cases involving Unification Church-linked organizations, arguing that the UC ties with the LDP have historically allowed it to cover up financial crime. Tsukada seems to believe that the task of historians is principally to identify UC's tendency towards crime and abuse. In contrast, I would point to an essay by Levi McLaughlin as more indicative of the consensus viewpoint among religious historians. We can be critical of the organization's tactics, but we must include the viewpoints of ordinary members and acknowledge their agency.
McLaughlin has never written a history of the UC (he specializes in another powerful group, the center-right Soka Gakkai), and Tsukada's viewpoint is unopposed at the moment. To show you how much work needs to be done on the unresolved question of how UC became so powerful in Japan, let's do a little dive into primary sources surrounding its origins.
There is a mythological origin story of the Japanese UC circulating in English language journalistic materials, which runs like this:
The founder of the Japanese branch of Moon’s church was Sang-Ik Choi, who spent much of his youth living in Osaka, Japan. [...] Finally, after years of frustration he managed to ‘convert’ fifty leaders of the ultra-nationalist Nichiren Buddhist sect, Riossho Kosei Kai [sic] (Establishment of Righteousness Rebirth Association) [sic] in late 1962, and with their help the [Japanese UC] began to grow. [...] Eventually, the Japanese UC became the largest and wealthiest of all Moon’s national branches, and some of the vast funds raised by its members were transferred to the U.S. and used by the American UC in its pro-South Korea ‘influence’ operations.
Prior to late 1962, the [Japanese UC] was impoverished, extremely small, and struggling for its existence. Less than four years later, it had become powerful, highly-organised and well financed. How can one account for this transformation? Although the data doesn’t permit us to clarify all of the details, it would appear that ‘friends in high places’ again played a role in this turnabout, as they did in Korea after the 1961 coup. [...] According to Kaplan and Dubro, ‘[a]lthough most political arenas [in Japan] have their kuromaku, the term most often applies to those men on the right – usually the extreme right – who serve as a bridge between the yakuza-rightist underworld and the legitimate world of business and mainstream politics’.
Our first hint of kuromaku involvement in the development of the UC in Japan is that it was none other than Osami (a.k.a. Henri) Kuboki, an aide to Rissho Korei Kai [sic] president Nikkyo Niwano, and Kaichi Komiyama, the chief of the organization’s Religion Department, whose ‘interest’ in the fledgling [Japanese UC] led to the crucial 1962 ‘conversion’ of the fifty sect leaders to Unificationism. In addition to occupying an influential position within the Rissho Kosei Kai, Kuboki was apparently a ‘yakuza lieutenant’ of Yoshio Kodama, one of the two most powerful kuromaku in postwar Japan.
source: Jeffrey Bale, 'Privatising' covert action: the case of the Unification Church, Lobster magazine, 1991
Basically, it's claimed that the UC in Japan was created artificially by the Korean CIA (KCIA) through their partners in the Japanese yakuza. The author argues that the UC would not have succeeded without a "mass conversion" of fifty "ultranationalist" Buddhist "leaders", who presumably had some secret instructions from the KCIA. This story has been repeated in several English language sources over the years, none of which seem to be able to spell "Risshō Kōsei Kai" correctly.
This story originates in the 1970s Koreagate scandal, in which South Korea was accused of influencing American Democrats in Congress. It's a sort of horror story which was spread inside the US as part of our internal political disputes. To believe this, you have to think that the UC is itself a conspiracy of dozens of far-right puppet masters manipulating foolish, uneducated people who didn't understand their religion was a bunch of lies. However, the story told by the founder of the UC's mission to Japan, Sang-Ik Choi, was quite different:
A telegram came from Seoul ordering me to return there. "Stop," it said. I wanted to die. After having sworn to go, and departed, how could I return to Seoul? It would be better to die. I sent a letter and a telegram asking them to wait three more days, and I hurried up my departure. The command was from Master, how could I disobey it? But I had resolved to go and departed; I could not return. . . . No matter what, I vowed to go, and I am determined. Even if I die in fulfilling my vow, I must go because I believe that I am going to save the Japanese and all of the peoples of the world.
This story passes the criterion of embarrassment (believer disobeying Moon), so it's reasonable to believe there was a reluctance by Sun Myung Moon to expand into Japan at all. This was actually the third attempt to preach in Japan, the previous two attempts having failed and reflected badly on Moon. The linked article also shows that Choi was deported back to South Korea in 1964, which is evidence against him benefiting from any "bridge between the yakuza-rightist underworld and the legitimate world of business and mainstream politics". This is why getting a believer's perspective can be very important for telling religious history accurately, even if you dislike the religion.
How about Osami Kuboki, the "yakuza lieutenant" from Risshō Kōsei Kai? He patiently attended a lecture by one of the Japanese followers who had never preached before. Here's how that follower describes those early days:
(continued)