First of all, I will readily agree with you that the Bible does not spell out in any degree of detail what His sacrifice involved. Common sense tells me that it involved more than merely dying. Jesus wasn't the only individual on earth who was ever crucified. As horrible a means of death as it undoubtedly was, it was a relatively common way by which criminals (particularly non-Roman criminals) were put to death under Roman rule. Whether Jesus had been crucified or had died of old age, His death was not -- in and of itself -- sufficient to accomplish what His Father required of Him, which was to redeem the souls of the entire human race.
In Isaiah 1:18 we read, "Come now, and let us reason together, saith the Lord: though your sins be as scarlet, they shall be as white as snow; though they be red like crimson, they shall be as wool." God requires that we be free from sin in order to enter into His presence. It's not a matter of whether own good deeds ultimately outweigh our bad deeds. Once we have committed even one relatively insignificant sin, we are no longer in a position to be able to stand before Him in innocence. We can and need to repent, but the damage has been done. Our record is no longer clean. The only way we can become perfect again is through the atonement of our Savior.
When we sin, we incur a debt which we are not in a position to be able to repay. The only one good enough to pay the price for our sins was Jesus Christ. Again, from Isaiah, we are told, "Surely he hath borne our griefs, and carried our sorrows: yet we did esteem his stricken, smitten of God, and afflicated. But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed." (chapter 53, verses 4-5). Payment for sin always requires recognition of the sin, followed by deep and sincere remorse. It involves the acceptance of guilt and the horrible mental anguish that results from this guilt. Jesus was not only punished for our sins, He assumed our guilt.
When we feel that we aren't worth saving, that we've sunk too low for the Atonement to be of any benefit to us, we underestimate its power. I believe the Savior descended below all things. He knows -- from His own experience -- how it feels to be where we are. 2 Corinthians 5:21 says that "...he hath made him to be sin for us, who knew no sin; that we might be made the righteousness of God in him." Taking upon Himself the sins of the world was painful. It was more painful than the crucifixion itself.