When these studies talk about the value of "close" relationships, they tend to point to properties like respect, trust, and emotional intimacy. Basically the relationship characteristics that someone with a secure attachment style tends to develop. (Which raises an interesting question, are they really finding that attachment style is the biggest contributor to happiness?)
It's not necessarily a function of how much time you spend with the counterparties, or exactly how many relationships you have.
If you have strong, healthy relationships with your kid, ex, and a friend or two you're probably already in the "good" column.