Secure attachment might not necessarily have a direct cause and effect relationship with better well-being later in life because there are other factors that can influence both characteristics. For instance, if a child had a particularly sensitive mother or father, this could be responsible for both secure attachment and better well-being. However, the secure attachment can not be said to cause the better well-being because in this case the sensitive caregiver is the reason for both.
I agree that there is no way to possibly generalize and form a conclusion on this debate. There are definitely many factors that are at play, and many of them intertwine for being possible effects for either of these situations. What we are able to discuss is the things that we know are causes for one or both of these things, even if they are not isolated.
Well another aspect to consider is over-attachment. A child could be over-attached to his/her mother, so that the "real world" possible causes this person to suffer a breakdown (or something) when they realize that the "real world" isn't as caring as their mothers were. I'm sure we've all heard the term "sheltered" used to describe someone.
I think sensitive in this case can be classified in being a secure caregiver. Secure caregivers teach the infants important characteristics such as love, understanding, and sensitivity as well. The term sensitive caregiver is very broad in this sense with many terms and definitions. But using this broad terminology for "secure" caregiver and having sensitive in that term it would seem that a secure caregiver would produce the outcomes of secure attachment and a good outcome.