It's not some one-for-one trade where Mom A shows up at Family B's home for her shift as nanny, while Mom B does the reverse.

In reality Mom A is a nurse, and cares for 20+ people in a day, and Mom B is a school administrator, both provide more than one-day-of-mothering value per day worked, so it's actually in everyone best interest that they leave their kids at daycare, consuming 1/6th of a less-talented person's day, enabling their professional output. Even the daycare worker multiplies their output by watching multiple children.