We've added no density? The population of the city went from 500k to 4 million since that sign went up. Koreatown has over 50 thousand people living per square mile. Is there an imbalance between jobs and housing today? Yes, but to imply that no housing has been built since the 1920s or that the entire area is low density suburban sprawl is a whole lot of hyperbole. Here is the population density map for reference:
https://upload.wikimedia.org/wikipedia/commons/e/e5/LACounty...