Fat chance

Why should that not also include trusting him to do what is right and just with all those people for whom you are concerned?

Well for starters:

John 14:6 (King James Version)

Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.

So clearly in this life the person born in downtown Baghdad -- who is very likely to become a Muslim let's face it -- has little chance of coming to God.

that upon death one comes before Jesus, and at that point they have a final opportunity to reject or accept him.

Well that's the answer then. The Muslim dies, he's standing in front of Jesus, looking around for any sign of Mohammed and not finding him. He just sees Jesus before him. In that case, if the Muslim still rejects Jesus then he goes to eternal damnation. Alternatively if the Muslim does accept Jesus then all the mosque-work done throughout the life of the Muslim is put aside with an "ok you were unlucky to be born in downtown Baghdad but at least in the end you made the right decision, you may enter".