First, it should be noted that this argument only applies to God as described in the New Testament. Second, the argument as described by Saige is missing a part. In full, it is:
  1. God wants to save everyone from damnation.
  2. The only way for a person to be saved from damnation is by believing in God/Jesus.
  3. God is omnipotent.
  4. Thus, God could make everyone believe.
  5. But not everyone believes.
  6. Therefore, God does not exist.
A solution to this is offered by Calvinist theology, although addressing this problem was not a motivation in the formation of their theology. The solution is to give a different interpretation to step #1. According to Calvinist theology, when the New Testament says that God wants to save everyone from damnation, it doesn't mean literally everyone, but rather everyone that God has already decided that he's going to save, the Elect that God chose for salvation before the beginning of time; God just doesn't care about saving anyone else from damnation. Humans are by nature so evil (see original sin) that he'b be perfectly justified in just letting everyone go to hell, so it's amazing that he's even saving the Elect.

Thus, the existence of people who don't believe in God is not a contradiction.