While I agree that what we might call "traditional slice-of-life" shows are rare today, they do still exist. The most recent example is
Showa Monogatari. It portrays a lower-middle-class Japanese family during 1964, the year the Olympics came to Tokyo. The family faces its share of issues -- financial problems, a difficult teen-aged daughter, a son who does not want to take up his father's machine-tool business, etc., etc. There is nothing earth-shatteringly dramatic about any of these conflicts; they are just the sorts of issues ordinary people face as they try to live their lives. This type of show would be played for laughs as a situation comedy in the US, but the Japanese treat it seriously. I wouldn't call it high drama, though, because the issues are not "big" enough. Nor is it a working-class tragedy like the plays Arthur Miller wrote after the War.
Showa Monogatari is just, well, "slice-of-life."
I'll readily admit that shows like these are not common, but they are distinctive and, in my admittedly constrained view of world popular culture, distinctively Japanese.