Stoic doctrine states that we should detach ourselves from our emotions and only worry about what we can control. If we follow this school of thought we will not waste our time on things that are futile, but only worry about that things that we have the capacity to control ourselves. By doing this we will live a better life because we will no longer waste time on what is out of our control, and therefore we will use our time concentrating on what we can, which will lead to better use of our time and will lead to a fulfilled life; not a life that we feel has passed us by due to concentrating on futile ventures.
while I do agree with the idea of not trying to control things that are out of our control. I do not agree with the view of apathy. if the world all held this view then it would be a cold place. feeling of pain and pleasure are what makes us human. pain helps us to be able to enjoy pleasure that much more. Caring about loved ones while in pain is another thing that makes us human and see no reason for apathy its part of life.
Yes, caring for others is a part of life, but trying to make others care for you is what the Stoics would argue against. You can personally control who you care for and whom you don't. But, you ca not control whether that person cares back. So you can care for whomever you would like, and do things that would earn their affection, but if the do not care for you, you cannot change that, and any minute spent trying to change their judgement of you, is a wasted minute.
You should accept that death is a part of life, and be prepared for not only yours but those who are around you as well. Grieving over someones death will not bring them back, so there is no reason to waste your time grieving. Instead do what you can to be prepared for their death and accept it when the time comes. By doing this you will not only make it easier on yourself when your wife or child does die, which they will, but you will also be able to move to productive actions when the time does come, in turn repairing you for your own inevitable death.
i have grived the loss of loved ones also been productive. being productive and grieving loss has nothing to do with each other. i gave a class presentation one hour after my father died, but when time permitted i did grieve his loss.