I would definitely challenge that assertion unless there was an actual study that proved this to be the case. That being said, I am no yakuza expert! So take everything I say with a grain of salt, really much like how you should read all of my posts!

Several yakuza affiliated corporations own large swaths of forest in Japan but certainly not the majority, which is mostly administrated by the government, at least in part. Many of these nominally government areas are worked on by government sanctioned companies. So what do the yakuza do with their share of the forests?

Well, for decades the yakuza have used many forests as illegal dumping sites for chemical, biological, and electronic waste, sometimes from their own companies and sometimes as a 'lease' to other corporations who would pay for the service (the appeal is that it's cheaper to illegaly dump the waste than to pay for the expensive procedures to properly dispose of it, particularly medical and electronic waste). For example, in 1996 a company with yakuza affiliations dumped medical waste in Nagano prefecture and it was a major scandal when it was discovered by several relatively 'famous' environmentalists.

If anything, many forests are aggressively cut by yakuza affiliated companies because they can be assured that they can avoid some government oversight by environmental agencies as well as interest groups, such as the Nature Conservation Society Japan or World Wildlife Fund Japan, due to their apparently deep ties with Japanese politicians.

Many yakuza companies have deep ties with government agencies, meaning they are able to secure claims to forests and other parts of land through political clout. This in turn allows to them to use them for legitamate business, such as supplying construction, which is the stereotypical yakuza economic venture, or actual land development. Theoretically, your friend's claim is plausible. Yakuza affiliates could theoretically just be hedging the market.

But many companies cut lumber in Japan and of course Japan imports plenty. I don't see how the yakuza could have cornered the wood market, whether decades ago or even today. The yakuza have incredible amounts of political influence for an openly criminal organization but they definitely don't dictate trade agreements. Natural resources used in construction, particularly wood, are certainly marked up in Japan but it isn't as if yakuza construction companies have anything even approaching monopoly.

The reason why we would know right away is that yakuza companies are incredibly open. For example, you can find the offices of the Yamaguchi group (the largest yakuza group today) very easily, since they proudly show their name on plaques out front. During sakura season, you can sometimes see areas reserved for Yamaguchi group members, clearly demarcated by paint and tarps on the ground. They even hold press conferences.

All their legitimate enterprises are run right out in the open, as they have been for decades. This is due to the fact that it is not criminal to be a part of a yakuza syndicate as well as the fact that many yakuza groups have implicit agreements with various law enforcement agencies in Japan.

As deeply entrenched the yakuza are, especially with the relatively increase in police crack downs compared to the 1990s and 1980s, I think it would have made the news if the yakuza held a monopoly in major natural resource production, especially if they were artificially raising prices on wood, which would directly impact the construction of new houses, which occurs INSANELY often in Japan because of the major differences in how homes are always depreciating assets in the Japanese real estate market, completely different from the US market.

Fun fact, because of the incredible rate of new houses and homes that have been built in Japan for decades, it has had a very vibrant and active architecture community, hence the very advante garde houses and interesting, sometimes almost blatantly experimental, designs that pop up. Such as this or this and of course these. It certainly helps that Japanese construction is priced astronomically so people who can afford to buy land and build a house are already in the income bracket that gives them leeway to let an architect go wild.

Jared Diamond book "Collapse" has a section about the very successful reforestation programme dictated by the Japanse government of the Tokugawa era.

So I say would say that the assertion that the preservation was thanks to the Yakuza is very shaky.

Some excerpts:

The 1657 Meireki fire, and the resulting demand for timber to rebuild Japan's capital, served as a wake-up call exposing the country's growing scarcity of timber and other resources at a time when its population, especially its urban population, had been growing rapidly. That might have led to an Easter Island-like catastrophe. Instead, over the course of the next two centuries Japan gradually achieved a stable population and much more nearly sustainable resource consumption rates. The shift was led from the top by successive shoguns

[...]

a full solution to the crisis required positive measures to produce more trees, as well as to protect land from erosion. Those measures began already in the 1600s with Japan's development of a detailed body of scientific knowledge about silviculture. Foresters employed both by the government and by private merchants observed, experimented, and published their findings in an outpouring of silvi-cultural journals and manuals, exemplified by the first of Japan's great silvicultural treatises, the Nogyo zensho of 1697 by Miyazaki Antei. There, you will find instructions for how best to gather, extract, dry, store, and prepare seeds; how to prepare a seedbed by cleaning, fertilizing, pulverizing, and stirring it; how to soak seeds before sowing them; (etc.)

While Diamond is not himself an authority on the history of Japan (and this and other of his books have been criticized for a number of reasons) the bibliography of Collapse contain a number of references to books on this matter, mainly three books by Conrad Totman: The Green Archipelago: Forestry in Preindustrial Japan (Berkeley: University of California Press, 1989); Early Modern Japan (Berkeley: University of California Press, 1993); and The Lumber Industry in Early Modern Japan (Honolulu: University of Hawaii Press, 1995).

I ask pardon as my post is mainly a cut and paste from an external source. But I feel that it answer pretty conclusively the original question: the preservation of Japan forest was a quite explicit government goal, not the byproduct of activities by criminal organizations.