Death and Life do not necessarily imply existence, unless you can show that existence is inconceivable without life.
Does a rock have life? As we know it, A rock is neither alive nor dead, it just is. It exists as distinct and separate from life as we understand it and therefore this suggests that existence itself is conceptually separate from life and death. Ergo existence cannot imply the concept of death, life, immortality or anything else. This suggests that existence just is.
However, what if our most basic assumptions about life are incorrect? What if life isn't cellular, or based around an ability to directly interact with and/or respond to surroundings? What if the assumption is that molecules and atoms are alive because they can display movement and state change, even if only sub-atomically? If these statements hold true, then does this necessarily invalidate the assumption that existence cannot imply death? Arguably again no if your meaning of death is that something was living and can cease to live.
What about if we change our frame of reference and imply that death isn't the opposite of life, but instead that death is the cessation of something, I.e.: The end of it's existence? If a building is constructed and exists for many years before earthquake destroys the building and turns it into rubble. Is this cessation of the building's existence the death of the building?
If you want to throw God into the mix, then what is your frame of reference? If you believe God exists, than does God exist? If you cease to believe God exists, then does God die, or is it that God never existed in the first place, or is it that God simply ceases to exist? Does existence rely on your ability to observe something existing, or does existence require a belief in existence?
Back to your question, when philosophers attempt to rationalize the existence of a divine entity, they are not questioning existence itself, but the existence of something specific. They are not really arguing if there is life or death or even if God is alive or dead or can live or die, but whether God exists at all. From this point of view, rationalizing the existence of God does not imply that one is attempting to rationalize existence vs death.