I think that attachment definitely plays a vital role in a child's development of a better well being later in life. The connection between a child and some kind of parent like figure is the very first relationship that any person has. They will compare all their future relationships to this. If a child has secure attachment they are much more likely to be able to develop trusting and functional relationships with others. On the other hand if they were never able to form that bond with a parental figure, and learned not to be able to trust them for protection or rely on them for other necessities, it would be really hard for that child to be able to trust and rely on others. This attachment plays a very important role in life, but so does environment. Having secure attachment with your parents does not guarantee a better well being, in fact all of your relationships with other people affect how you will interact and perceive the future people that come into you life. If you learn not to trust a large enough amount of people, it may be hard to trust people you meet in the future, despite the secure attachment you had as a child with your parents.
