Maybe. But I'll counter with my own anecdote. I raised my daughter as a single working dad. No mother in the home at all. And yet, I got in plenty of volunteering and socializing anyway.
My dad spent most of his time on the road, as a travelling salesman. I did spend more time with him as life went on, however. My dad never volunteered though, and all of his social connections were former high school friends that he saw once or twice a year.
My dad did a heroic job, too. I’ll always be grateful for him raising me. I still wish I had had a mom at home though, like most of my friends did.
However, when you shift the majority of connections to those made through the workplace, you end up with just a bunch of connections running roughly parallel to each other. Not so much a fabric as a pile of string.
Probably many flaws to this analogy, but that is basically the idea of "third places". That you need somewhere where people interact based on something different from what they do for a living or who their direct family members are.
A lot of the mothers were. Actually we only had 6 teachers, for 6 classes/grades. No other personel ( one of the teachers was the 'head' ).
Even the simple decline in people who identify their neighbour as a friend can be explained by a lack of women living and socializing in the neighbourhood during the daytime. The increased use of paid childcare would similarly reduce the amount of children living and playing in the neighbourhood during the day.
Overall, this would contribute to the feeling that the suburbs are ghost towns during the day, and families arriving home from school/work would rush inside to eat dinner before chauffeuring the kids to all of their extracurriculars.