I see "personal life" as much larger than "childcare with friends". Can your wife go mountain biking? Can she attend a lecture that coincides with nap time? Can she delve into a tough book?
Those things are all easier to do as a stay at home parent than if both parents are working, because then you end up dedicating all of your non-work hours to family, chores and errands anyway.
This seems a silly question. During the workday, neither spouse can go mountain biking. Outside of the workday both spouses can go mountain biking. This is true whether one or both spouses are wage-earners.