I understand and agree with your position that Jesus is the only way we can be saved, as no one is capable of keeping the law. My question is: Does Jesus, through his atoning sacrifice, save those who are ignorant of his existence through some mechanism that I (and apparently many others) am unaware of?
The Bible tells us the following things about God.
1. He is just and can do no wrong (Deut. 32:4) This means he cannot condemn any one due to ignorance or intellectual incapacity and neither can he allow any guilty party to escape punishment. Children and the mentally handicapped are exempt due to the above but all others are accountable (Romans 7:9).
2. He has made Himself so obvious in the creation that no one can claim to be ignorant about His existence (Romans 1:20)
3. All men have sinned and fall short of the glory of God (Romans 3:23) Everyone is guilty and accountable for their sins.
4. He doesn’t want anyone to perish but for all to come to repentance (1 Tim 2:3-4, 2 Peter 3:9) His desire is for everyone to be saved.
5. He loves the world so much that He gave His only son, that whoever believes in Him will have everlasting life (John 3:16). This is the remedy He provided for our sin problem. We have to ask for it (Romans 10:13) but everyone who asks will receive it unconditionally (Matt. 7:7-8).
6. The death of His Son is the only remedy God has provided for our sins (John 14:6).
7. Through the Son’s death the Father has made peace with His creation (Colossians 1:19-20). By accepting the Lord’s death as payment for our sins, we have been reconciled to God as if we never sinned in the first place (2 Cor. 5:17).
8. Man is destined to die once and after that to face judgment (Hebrews 9:27)
When you put all these things together, the only conclusion you can draw is that everyone ever born has to receive at least one bonafide offer of salvation by the blood of Jesus during his or her lifetime. There can be no exceptions. Whether we understand how God has accomplished this or not is irrelevant. His word, His reputation, and His character are at stake and depend on Him doing it.