As already noted, if you take away half of the existing workers, of course the economy would collapse, but that's not special to men. If every existing male worker were magically replaced with a woman, things would go on as usual. We don’t need men to do those jobs, we need somebody to do those jobs. In fact, the reverse would be much worse, as they found out in Iceland because even though most women work outside the home, they still do the vast bulk of home and child care as well.

https://www.bbc.com/news/magazine-34602822

Instead of going to the office, doing housework or childcare they took to the streets in their thousands to rally for equal rights with men. It is known in Iceland as the Women's Day Off, and Vigdis sees it as a watershed moment. "What happened that day was the first step for women's emancipation in Iceland," she says. "It completely paralysed the country and opened the eyes of many men."

And besides all this, the phrase “I don’t need a man” is not about practicalities, it’s a reaction to the pervasive messaging that still exists to some extent that a woman is not a real adult unless she is married and has children. It’s a rejection of that.