I am glad to see you do not embrace a "Universal Salvation" theology. You are also correct in that there is a "universal fair chance" at salvation. However, that universal fair chance occurs in this life only! Not after this life is over.
Everybody is given a chance during this life to receive the salvation mentioned in the Bible. Once we die there is no more chance. God provides us with every opportunity to accept His gift. It is up to us to recognize it and to take Him up on His offer.
When Jesus died, He did so willingly to buy us back and to redeem us from the bondage of sin. The only way to receive salvation is to accept the gift He made available through His death on the cross. There is no other. At times, I wish there was, but this is the way it has to be -- regardless of what we may wish or think.
We can think that God is "unfair" by only permitting one way to heaven -- but how much more fair can He be but to open it up to everyone? Also, we are in no position to judge the God of the universe. He, after all, is the one who will rightfully judge us.
We, as humanity, are in this sinful state by our own fault. We, that is you, me, and the rest of humanity choose to sin on our own accord. But God, through the maifestation of His love for us, provided a way out! Out of love, He came to earth Himself, as a man, to pay the penalty for our sin! All we have to do is accept that gift! How could He make it any more fair, or any more easy?
God IS so good!