Hood never said that the Chief was "dead," he said that he was "gone." He was either saying that he thought the Chief was dead, or he knew he wasn't dead, just gone.
What would be best is beating the game again with subtitles on, starting of course from the last checkpoint of the last level to make things easier.
We never saw the ship crash. We don't know he dies when it does. In fact, we don't even know he crashes; he could get blown up by another ship, or a big rock hits the ship and sends him in a different direction, etc.