Not quite sure how old age fits the riddle. When talking about old age, wouldn't coming to it and awaiting it be the same thing? Given that, it can't have two different outcomes, one of them being life.
If I were to argue you're point, I would say the opposite of the riddle would be true. If one awaited old age, then death would come faster because of all the worrying and stress over getting old and dying. The best way to live a long time is to simply accept getting old and not await anything in a pessimistic light.