I question the accuracy of that report. Sociologists are well aware of the fact that responses to questionnaires are strongly influenced by the ways in which the questions are framed. My educated guess is that the research for this report was done in a very sloppy fashion.
There's a secondary issue here. Japanese birthrates are well below replacement levels, as is the case in some Western European countries. And the inversion of the age pyramid is putting a big strain on their social security systems.
Aside from Denmark, Western European countries are using immigration as a financial shock-absorber. However there's a trade-off. Immigration is also contributing to a Clash of Civilizations within some European countries.
On the other hand, the Japanese have made a very conscious decision that immigration is not going to be the solution to their social security funding problems. Considering their current economic situation, that's very reasonable.
Extinction? Hardly. My wild guess is that the Japanese birthrate will pick up a bit when the population density has thinned out to a more comfortable level, when there's room for most Japanese people to live in less-cramped apartments, when there's less air pollution, and when long-term food security is less of a worry.