The article doesn't mention offering people money to leave. The Japanese invested in transit and built more housing to bring down prices. Japan is also a very small country. Housing prices are determined by supply and demand. You can't do much about demand, but you can add supply.
By the way, rent controls just make the situation worse, by reducing incentives to build new housing.
What does economic evidence tell us about the effects of rent control?
Here's the conclusion for those who don't care to click the link:
Rent control appears to help affordability in the short run for current tenants, but in the long-run decreases affordability, fuels gentrification, and creates negative externalities on the surrounding neighborhood. These results highlight that forcing landlords to provide insurance to tenants against rent increases can ultimately be counterproductive. If society desires to provide social insurance against rent increases, it may be less distortionary to offer this subsidy in the form of a government subsidy or tax credit. This would remove landlords’ incentives to decrease the housing supply and could provide households with the insurance they desire. A point of future research would be to design an optimal social insurance program to insure renters against large rent increases.
Here's the full text of the NY Times piece:
Yuta Yamasaki and his wife moved from southern Japan to Tokyo a decade ago because job prospects were better in the big city. They now have three sons — ages 10, 8 and 6 — and they are looking for a larger place to live. But Mr. Yamasaki, who runs a gelato shop, and his wife, a child-care worker, aren’t planning to move far. They are confident they can find an affordable three-bedroom apartment in their own neighborhood.
As housing prices have soared in major cities across the United States and throughout much of the developed world, it has become normal for people to move away from the places with the strongest economies and best jobs because those places are unaffordable. Prosperous cities increasingly operate like private clubs, auctioning off a limited number of homes to the highest bidders.
Tokyo is different.
In the past half century, by investing in transit and allowing development, the city has added more housing units than the total number of units in New York City. It has remained affordable by becoming the world’s largest city. It has become the world’s largest city by remaining affordable.
Two full-time workers earning Tokyo’s minimum wage can comfortably afford the average rent for a two-bedroom apartment in six of the city’s 23 wards. By contrast, two people working minimum-wage jobs cannot afford the average rent for a two-bedroom apartment in any of the 23 counties in the New York metropolitan area.
Maintaining an abundance of affordable housing has its downsides. Green space is scarce in Tokyo, living spaces are small by Western standards, and relentless redevelopment disrupts communities.
But the benefits are profound. Those who want to live in Tokyo generally can afford to do so. There is little homelessness here. The city remains economically diverse, preserving broad access to urban amenities and opportunities. And because rent consumes a smaller share of income, people have more money for other things — or they can get by on smaller salaries — which helps to preserve the city’s vibrant fabric of small restaurants, businesses and craft workshops.
As political leaders in urban areas around the developed world grapple with how best to revive their cities in the aftermath of the pandemic, Tokyo offers a template.