This is getting us onto the key point - what is belief and faith all about? What does God give to everyone and what happens to those with faith? However you don't seem to follow up on this and explain further what the difference is between the believer and non-believer. Perhaps you could answer your own question: "what *is* belief?"
The only difference between a believer and a non believer is the point and time in which God has drawn a person to meet Jesus. (john 6:44) That honestly may be oversimplifying it, but that is the basic thing I see in a nutshell.
What is belief? Belief is a result of an event. People think they choose to believe, but belief is different than thinking something might be, something could be, etc.
Belief in Christ is only a result of meeting him. This is why there is churches full of people who say they believe, but in their heart, really do not.
If you taste a bite of food, you cannot believe the food to be good until you taste it. You might consider that it is good, you might wonder if it will be good, you might hear everyone else raving about how good it is, but until the event of the food hitting your taste buds and the time taken to experience the after taste, there is no way, YOU can truly believe that it is good.
You can even try to tell yourself, "its not that bad" "ah, it's pretty good" "oh, it's ok" but the event has already taken place and despite what you try to convince yourself of, you will either truly know and believe its good, or truly know that you do not like it. But the event will have manifested the belief, not your choice to do so or not. This is why self delusion is so hard to spot, many people will argue that belief is a matter of your choice, it is not.
Ro 1:20 For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, [even] his eternal power and Godhead; so that they are without excuse:
Christian often use the passage that this verse is in as a key to say that when they are judged God made it known to them that he was real sometime in their life. And if they rejected that knowledge then they have no excuse when it comes time to face going to an eternal hell.
But what it is telling us is that, there is no one who will be defenseless to "UNDERSTAND" this event right here.Phl 2:10 That at the name of Jesus every knee should bow, of [things] in heaven, and [things] in earth, and [things] under the earth;
Phl 2:11 And [that] every tongue should confess that Jesus Christ [is] Lord, to the glory of God the Father.
The whole thing of salvation is a process, the faith God gives every man, the things he shows us in our lives, so that when we meet Christ there is a foundation made so that the only thing that can happen when we meet Christ is belief in Christ.