Even Sommers doesn't try to claim the gap dissappears when you account for issues like hours worked

What are you talking about? That's exactly what she says https://www.youtube.com/watch?v=58arQIr882w

...she has absolutely no rationale for why women, if they are the free economic agents she claims, would "choose" to be underemployed.

Actually it's quite obvious.

Labor force participation varies by marital status and differs between women and men. Divorced women had a higher labor force participation rate than married women, 64.2 percent compared with 58.4 percent. By contrast, married men were more likely to participate in the labor force (73.5 percent) than divorced men (66.1 percent). (See table 4.) Among mothers, the labor force participation rate was higher for those with children 6 to 17 years old than for those with younger children. In March 2014, the rate for women with children 6 to 17 years old was 75.8 percent. The rate for those with children under 6 years old was 64.3 percent, and the rate for women with children under 3 years old was lower, at 61.8 percent. (See table 5; data were collected in the 2014 Annual Social and Economic Supplement to the CPS.) Unmarried mothers have higher labor force participation rates than married mothers. In March 2014, 76.2 percent of unmarried mothers with children under 18 years old were in the labor force, compared with 68.4 percent of married mothers with children in the same age range. (See table 6; data were collected in the 2014 Annual Social and Economic Supplement to the CPS.)

The largest driver of the 'wage gap' is familial status. Women are more likely to drop out of the work force, work less hours, or work in more flexible fields, primarily due to having children.