Children that have secure attachment are more outgoing and more likely to approach strangers. This behavior allows them to develop more contact with people and develop better social skills. Having better people-skills allows on to be able to handle relationships better even with other types of attachment styles. Thus, it is contact with other people that causes a better well-being in later life.
I don't think that's true all the time. For instance, I am more of an introvert when I meet new people in a new environment. But, I am definitely sure that I have secure attachments with my parents, families, and old friends. Even though I might not be friends with everyone, I value my relationships and care a lot like how my parents and families cared about me.
I think that that is true in some particular cases. Certainly developing better social skills allows one to succeed in the ways you mentioned, but I don't think that it is always necessary. Individuals can maintain relatively successful relationships with particular people although they may be considered lacking in social skills. Different pairs of people interact in different ways, and the social skills required for each variety of interaction differ across social groups.