His mother changed their family name when he was a child. He was "outed" during his first election campaign, but this did him no appreciable harm, and the media made a bigger deal of his father having apparently been a low-level Yakuza gangster.

Also, it's difficult to disentangle "extremely hesitant to speak" and "genuinely unaware/doesn't care": as far as I can tell, for most Japanese, the issue is an irrelevant historical relic. Living in Tokyo for years, the only reference I ever heard or saw a single piece of train station graffiti proclaiming that a certain train station was an etamura (derogatory term for buraku village). As noted, around Osaka it's a bigger thing, but the days of employers or prospective inlaws routinely engaging detective agencies to root out any trace of buraku origin seem to be long past. (Although doubtless you could still do this if you really wanted to.)

Finally, it's indeed a political issue, and often in unsavoury ways: the article you link to goes into some detail about how the Buraku Liberation League, which has... non-mainstream views on how many buraku there are and how badly they're persecuted, has Yakuza ties and earns considerable profits from government assistance projects.