We read in the Doctrine and Covenants, "Behold, there are many called, but few are chosen. And why are they not chosen? Because their hearts are set so much upon the things of this world, and aspire to the honors of men, that they do not learn this one lesson—That the rights of the priesthood are inseparably connected with the powers of heaven, and that the powers of heaven cannot be controlled nor handled only upon the principles of righteousness" (D&C 121:34-36). Fore-ordination does not have anything to do with who will be able to return to live with Heavenly Father again, unlike the Calvinist doctrine of predestination. Fore-ordination and predestination are similar but they do differ by degrees.
The LDS doctrine of salvation is quite liberal, especially compared to the Calvinism of the Puritans - however, it's not our doctrine, it's God's. That is, Mormons did not create the doctrine, God did. We believe that all who have ever lived upon the earth will have the opportunity to receive and accept the ordinances required to return to live again with our Father in Heaven. All people will eventually have the necessary ordinances performed for them but not all will accept (or possibly even be able to accept based on their lives here) the ordinances. We believe that all children who died before the age of accountability (age 8) will return to live with God again - this is because little children are whole and incapable of committing sin (see Moroni 8:8; Mosiah 15:25) because the Savior's Atonement freed them from sin; they are declared blameless by God through the power and grace and mercy of the Atonement.
That is radically different from Puritan beliefs about children; they thought children were full of sin and mischief and evil. Children, to the Puritans, had to be "broken." LDS doctrine holds children to be inherently good. Christ also taught we should become as little children (see Matt. 18:3). While LDS doctrine about salvation is liberal - we believe that most will receive some degree of glory (they at least kept their first estate and chose in the pre-mortal realm to follow our Father's plan of salvation by coming to earth) but not all will live with God forever more - LDS doctrine is not as liberal as some other Christian faiths, such as Universalists who believe that all will be saved; God's power is so great that He will bring all humankind back to Him. On a continuum, LDS doctrine is between the Calvinists and the Universalists.
Calvinism in New England espoused "Five Points" of doctrine, similar conceptually to our thirteen Articles of Faith (i.e., these Five Points are some of the basic and distinctive Puritan doctrines just like the LDS Articles of Faith cover the basic doctrines of Mormonism). These Five Points were: " the natural condition of humanity was total depravity,... salvation was beyond mortal striving,... grace was predestined only for a few,... most mortals were condemned to suffer eternal damnation, and  no earthy effort could save them" (p.112). That seems to be quite a depressing set of dogma! However, the Puritans did not live depressing lives - although they were austere in many ways - but they were never sure of their salvation. Their glasses of salvation were never more than half empty. They constantly sought and desired the mercy and grace of the Lord Jesus Christ, as we all should. Another quote will further establish the Puritan views of salvation. About one Puritan it is written, "Always before, when [Thomas] wept for his sins, he had kept some feeling of human merit. Now he knows he has none, that the natural man, even when seemingly a good man, is only a beautiful abomination, for the natural man has had no merit since Adam's disobedience, and Hell is his just destination" (Simpson as cited by Fischer, p.116).
But does not that passage sound a bit like King Benjamin? "For the natural man is an enemy to God, and has been from the fall of Adam, and will be, forever and ever, unless he yields to the enticings of the Holy Spirit, and putteth off the natural man and becometh a saint through the atonement of Christ the Lord, and becometh as a child, submissive, meek, humble, patient, full of love, willing to submit to all things which the Lord seeth fit to inflict upon him, even as a child doth submit to his father" (Mosiah 3:19). There are similarities between Puritanism and the first part of King Benjamin's statement but there is a key difference. King Benjamin stated that we need to become as a child in order to overcome the natural man!
Children are naturally good; the natural man is a learned and developed state, not an inherent state. We learn to be natural men and women! We all give in to the natural man at some point but we all start out good and clean and pure. I do have to point out that it is only through the atonement of Christ the Lord that we become saintly, or saved and sanctified. Again, as members of The Church of Jesus Christ of Latter-day Saints we believe that people are inherently good - the Puritans did not. The Savior's Atonement freed us from the transgression of Adam. Joseph Smith wrote, "We believe that man will be punished for his own sins, and not for Adam's transgression" (Second Article of Faith). I believe Joseph wrote that in part because it was different from the prevailing beliefs of many of the religions of his day.
Fischer, D. H. (1989). Albion's Seed: Four British Folkways in America. Oxford University Press, Oxford.