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?
Friends and family can help watch the kids- taking a bit of personal time is important too.
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.
Humorously, cycling with a child safety seat was absolutely something I did on a mountain bike while I was a full-time mom with small kids. I did it until I was like 8 months pregnant with my second child and my belly was too big for me to get on a mountain bike with child seat attached.
On the weekends, yes. My wife and I went skiing almost every weekend. Sometimes we take our daughters. Sometimes other family watched the kids. It's not a fair comparison, because I never go skiing during work. Although, my wife has discussed going skiing / camping with some of the moms during the weekdays when the kids are a bit older. So that's another win for moms I guess.
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.