To me, it means:
It is unreasonable to expect parents to raise well-rounded children without support.
In order to function intelligently in the world, a child needs to learn first-hand that people can be very different from one another, and that it doesn’t mean that a particular person is right or wrong/better or worse. This is done by safely (and graciously) growing up around people of different ways of life, age groups… and yes, religion/belief systems, etc. Seeing how different people treat each other, themselves, and how they react in certain situations teaches the child that they have a choice in how they live their lives, and they begin to practice making those choices in a safe environment.
If there are enough people to provide day-to-day support of a family with a child, there is less stress, less fear, less anger, less resentment and more joy, more gratitude, more learning, and more love to go around because experiences will vary, which always makes life more interesting, and provides something new to talk about at the dinner table, and so on.
A parent, when beginning to become stressed, or is too tired, or is dealing with some state of mind or other incapability that interferes with their ability to focus on the child can easily and quickly find a brief reprieve without having to resort to some electronic babysitter; and the child ends up gaining experience with interacting with another person, the care-giver has an opportunity to offer assistance and enjoy sharing time with a child, and the parent avoids frustration and burn-out.
It means that there are many people the parents trust with the life of the child because the child is truly important to everyone else, and they want to support the family.
It means that a child has a view of the world that is as realistic as possible to prepare them for adulthood when they will have to form and maintain relationships and perhaps raise children of their own.
This probably doesn’t answer all your questions exactly, but I imagine that if such a village existed, there would be an understanding of who was responsible for what, because it would largely depend on the specific abilities, availabilities, and extent to which an individual wanted to participate (even the parents – not every human who procreates knows everything about everything!). For example, some people would rather care for babies, when others would rather help with teaching a particular skill, or help them deal with emotions, or whatever.
This type of village means that there’s always someone to go to when help is needed, or some time is needed away from the family, but in a way that everyone knows that everyone is safe.
Unfortunately, this is not the norm in our society, if it’s even possible.
Fortunately, humans are very adaptive and resilient and end up being able to survive somehow. ;)