I believe the fact that securely attached children are more likely to have higher well-being later in life is inherited. When a child was first born, he couldn't already have formed the attachment style to his caregiver. It is the different ways parents foster their children that let the kids learn different types of attachment. How did the parents learn to foster their children? Of course they could just borrow fostering techniques from books, but it's more likely that they educate their descendants the same ways they were taught by their parents. Thus if the parents come from well-off families, they are more likely to spend time (they have fewer things to worry about) and make their children securely attached (after all, most parents want their children to behave normally). That results in the children growing up the same ways as their parents, and they will tend to be well-off in the future, just like their parents.
In many ways i agree with this statement. The economic status of families may correlate better with well being. Babies who are initially well off may be more inclined to be well off later. Because the parents have the proper resources to provide for thei child, the children who have good attachments may be raised better, more educated and have a higher well being later on.