When someone on their deathbed, in their last moments, begins to perceive Hell as more than just academic; they know it's coming and are likely to become very concerned about ending up there. In moments like that, people don't need religion, no, they need a knight in shining armor, so to speak, and it just so happens there's a God-given knight available.
● Luke 2:8-12 . .And there were shepherds living out in the fields nearby, keeping watch over their flocks at night. An angel of the Lord appeared to them, and the glory of the Lord shone around them, and they were terrified.
. . . But the angel said to them; "Don't be afraid. I bring you good news of great joy that will be for all the people. Today, in the town of David, a savior has been born to you; he is Messiah, the Lord."
The angel announced the birth of a savior; defined by Webster's as one who rescues. We've all seen examples-- lifeguards, firemen, cops, emergency medical teams, Coast Guard units, snow patrols, and mountain rescue teams. Rescue workers typically save people in distress who are facing imminent death and/or grave danger and utterly helpless to do anything about it.
In other words: Jesus Christ is a lifeline, so to speak, that God is all set to throw to anyone and everyone for whom destiny in Hell is a foregone conclusion if only they have the good sense to plead guilty and throw themselves on the mercy of the court by this simple, naive prayer:
"God, I know I'm a sinner. I would like to take advantage of your son's death."
Does Jesus' Father honor those kinds of prayers? Well if His son's story of the tax collector at Luke 18:9-14, and the account of the malefactor crucified along with Jesus at Luke 23:38-43 are truthful indicators; then I can honestly, and confidently, attest that He does, and He will.
NOTE: Just about everybody who's ever heard anything about Christianity is aware that Jesus Christ died for the sins of the world, but what is often unknown is that it was personal; as Isaiah 53:6 says: "The Lord has caused the iniquity of us all to fall on him."
In other words: the iniquity of each of us fell on him, i.e. any name we might pull out of a hat, and as many names as we might pull out of a hat: that one name, and each name, is an individual for whom Christ gave his life; there are no exceptions. It's no wonder then why the angel announced not just joy, rather, "great joy" that will be for all the people.