If you look at Glaeser's publication[0] (page 10) he defines the rate that he is citing as "the share of men who are not currently employed, or one minus the employment to population rate for prime men". That is a much broader measure of unemployment than even the already very broad U6 measure that NYT is reporting as this would include every single person without a job regardless of the reason that they don't have a job. This number would include full-time students, stay-at-home parents, disabled people, retirees and so on. For comparison, using that measure the absolute lowest level of unemployment that the US has ever had for the age 16+ population as a whole is about 35%[1] (this source reports employment to population rate, so it will be 100% minus the rate that Glaeser is using) or 18% for all people age 25-54[2].

Men in that age range are actually employed at a higher rate than most other demographics. He is making a point about men in that age range being employed at a relatively low rate compared to historic rates for that same demographic, not their employment rate compared to other demographics now. Men age 25-54 have at times had rates getting close to 5%[3].

However, this can't entirely be blamed on a poor job market because it has become much more acceptable for men to take the role of a stay-at-home parent or caretaker than it would have when their employment rate peaked in the late 1960s. There are almost far more students in that age range than there would have been in the 60s.

[0] https://scholar.harvard.edu/files/glaeser/files/jobs_for_the...

[1] http://www.multpl.com/us-employment-population-ratio

[2] https://fred.stlouisfed.org/series/LREM25TTUSM156S?utm_sourc...

[3] https://fred.stlouisfed.org/series/LREM25MAUSA156S