If one of my love ones were to die, I, too, would feel a deep sadness. However, if it were my mother or my grandmother (the two people in my life that I could simply not live without) I would be utterly inconsolable.
When a person speaks of their "deep sadness" or "deep sorrow", that is merely an expression of grief and is not indicative of their lack of feeling. Those are probably the words I would choose to describe my feeling for the majority of my friends and family, should I ever have the misfortune to lose them.