In your theory, those who had a sense of God but were pagans, worshipped idols, whatever....all could be saved as long as they had a sense that "god" would send a "redeemer" to give them eternal life.
That's exactly what I'm *not* saying. Note that I specifically included the word "faith" several times in my post; I didn't include the weak phrase "sense of God." Those with faith in the One whose name they do not yet know will be saved.
Those who worship idols are worshiping idols rather than the One True God. There were numerous people before Jesus' arrival who rejected God, and there are numerous people now who, though they don't know the name of Jesus, are rejecting him.
In the end, it is good to share the Good News of salvation with those who haven't heard it. Living this life with the knowledge of Christ is a blessing, and it's good to share that. And apart from Christ there is no salvation at all.
In the end, it's a matter of the heart. A heart surrendered to God, with faith in Him, is saved through Jesus.