God does indeed promise eternal life. John 3:16 is pretty obvious of that and such plain and simple verses probably explains why there are so few (probably not any) Christian traditions that deny eternal life is part of God's promise.
Generically, God promises restoration continually in the Bible (e.g Acts 3:19-21; Revelation 21:1-5; 1 Peter 5:10; 2 Corinthians 5:17 1). The very large majority of Christians believe that Eternal life was apparently part of our original state. Even the non-literalists, like Theistic Evolutionists, believe The Fall was something and that the story highlights that we were once in perfect communion with God. The Gospel is the promise of that restoration, the restoration of perfect communion with God and we get to be in communion forever.
Most Christians believe that the wicked will also live forever, if you can call eternal hellfire "living". This further completes the answer for those who believe in an eternal, indestructible soul. The promise is actually restoration of communion with God, not that you get to live forever, per se. However, this leaves Annihilationists with some further explaining to make and I'm not sure how that would make it.
So, I personally do not think the typical Christian sees the promise of eternal life as pandering to our "greed for life", but rather as part of the promise of restoration, leading to a belief that death is unnatural (at least for the Creationist, non-evolutionist variety), or more like "naturally undesirable". I recall reflecting on it many times saying "Death just seems wrong, like it shouldn't be part of our lives." A more naturalist approach, as I suspect you are more familiar with, would likely view death differently, and perhaps as even serving a greater purpose. Cynics have throughout the years, especially since the Enlightenment era, simply agreed. A promise of eternal life is pandering to human fear of death, but even if Christians agree, it doesn't change the fact that the promise is more accurately restoration which includes everlasting life.
- Old Testament restoration verses are also plentiful, but seemingly focus on immediate restoration, not lasting restoration. This highlights the Gospel message, rather than diminishes it, showing that with God (literally Emmanuel, God was with us) we have our eyes opened to see how great and powerful God truly is. He can not only restore us for today, but for all eternity.