I think a lot of the characteristics attributed to securely attached children are moreso reflections of their own genetics and innate personality than merely the relationship between infant and caregiver. Thus, the child's future well being relies more on its own self determination and less on the context in which it grew up.
I think theres a relationship between the genetics and innate personality and the relationship between infant and caregiver. The genetics aspect predisposes a child to act a certain way and it will be reflected in the relationship between infant and caregiver. Whether the attachment is secure or anxious has little to do with the caregiver (in the strange situation the scenario was the same for every mother/child combination). It was all about how the baby responded to that situation and his response is probably based on some sort of genetic influence.