Because, uh...Abe Shinzo's family has already got one, you see. (Link)
What is more, it is the only one in Japanese hands.
Abe Shinzo and his relatives have been prime minister for over 20 years out of the 68 since the Japanese surrender to Allied forces (more if you include the marriage relations - Link - J). Having Kishis as the holders of Japan's two Nobel Prizes for Peace would be beyond satire, especially in light of recent revelations of the flexibility of Sato Eisaku's anti-nuclear stand. (Link with key photo)
As for selling the concept of Abe's renouncing of Japan's claims on Dokdo/Takeshima by putting such a renunciation on a level with Sadat's acceptance of the Camp David accords and Nixon's opening with China, one could hardly fault Abe Shinzo for dryly noting, "And taking these bold leaps resulted in huge personal benefits for both men, yes?"
Not that there is not merit in renunciation -- for the United States. Dokdo is a flashpoint in South Korean/Japanese relations, complicating the establishment of an integrated regional security structure of U.S. allies -- not that the establishment of a Japan-South Korea axis in such a structure would in any way be perceived as the crucial step in a strategy of containment of China, much less be one.
Acquiescing to South Korea's military seizure or liberation (choose one) and garrisoning of Dokdo/Takeshima as a fait accompli would make it harder for Japan to hold any sort of line with Russia over the sovereignty of the Southern Kuriles/Northern Territories. It would also be very hard for the Prime Minister to explain to the families of the dozens of Shimane Prefecture fisherman injured or killed and thousands jailed during the period 1954 to 1965 -- from the arrival of South Korean forces to the normalization of relations (not that there is any chance that such a betrayal would lessen the Liberal Democratic Party's hammerlock on Shimane, mind you) -- unless he wrested from the South Korean government a sincere apology for the brutality it showed toward Japanese civilians.
A mental retort of "And it will be a cold day in hell before that ever happens" is not inappropriate here.
Having South Korean forces, armed and armored, occupying Dokdo/Takeshima, islets lacking fresh water sources, scanning the horizon for the Japanese invasion that will never come, and having patriotic South Koreans spending money, time and energy trumpeting South Korea's claim on Dokdo/Takeshima in venues appropriate and inappropriate all over the globe plays into the vital revisionist theme of Japan as victim, in this case of an irrational and therefore indissoluble South Korean paranoia.
In the end, my puerile association of the Clad/Manning appeal with the one King Arthur makes to the French soldier is not far wrong. Abe Shinzo, like Guy de Loimbard, has a stout fortress. An appeal to come outside to join an idealistic quest is something about which (and who could blame him) he is not very keen.