That seems to be more about facing your own death than someone else's. I see no solution to reducing the grief at losing someone other than to care about them less. If grief is the price I have to pay for loving in this life I am happy to pay it.
I do however agree that in the west we are too far removed from the reality of death and perhaps this is why we don't know how to deal with it when it happens, hence the stiff upper lip and tabooness of it all. I don't think being more in touch with mortality would magicly make grief go away but it might help us to deal with it more healthily - e.g. expressing our emotions, being willing to talk about death when it happens (and before) rather than tip-toeing around the subject or sweeping it under the carpet.