Our status as a covenant people is a lot more tenuous than we would like to believe. Paul taught plainly that status as children of the covenant God made with Abraham means that we have exercised faith in Jesus: we have obtained the word of Christ and we have lived the word of Christ.
That the blessing of Abraham might come on the Gentiles through Jesus Christ; that we might receive the promise of the Spirit through faith. For ye are all the children of God by faith in Christ Jesus. For as many of you as have been baptized into Christ have put on Christ. There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus. And if ye be Christ's, then are ye Abraham's seed, and heirs according to the promise. (Galatians 3:14, 26-29)
Nephi stresses this same teaching: covenant status is available to all who will repent:
For behold, I say unto you that as many of the Gentiles as will repent are the covenant people of the Lord; and as many of the Jews as will not repent shall be cast off; for the Lord covenanteth with none save it be with them that repent and believe in his Son, who is the Holy One of Israel. (2 Nephi 30:2)
And again:
Hearken, O ye Gentiles, and hear the words of Jesus Christ, the Son of the living God, which he hath commanded me that I should speak concerning you, for, behold he commandeth me that I should write, saying: Turn, all ye Gentiles, from your wicked ways; and repent of your evil doings, of your lyings and deceivings, and of your whoredoms, and of your secret abominations, and your idolatries, and of your murders, and your priestcrafts, and your envyings, and your strifes, and from all your wickedness and abominations, and come unto me, and be baptized in my name, that ye may receive a remission of your sins, and be filled with the Holy Ghost, that ye may be numbered with my people who are of the house of Israel. (3 Nephi 30:1-2)
Over and over we are taught how straightforward the process of becoming Israel is. However, we are also taught plainly that we can choose to reject our opportunity to become a covenant people. Paul tried to convince the Hebrews that their covenant status was not guaranteed to continue despite their wickedness. He said,
I say then, Hath God cast away his people? God forbid. For I also am an Israelite, of the seed of Abraham, of the tribe of Benjamin. God hath not cast away his people which he foreknew. Wot ye not what the scripture saith of Elias? how he maketh intercession to God against Israel, saying, Lord, they have killed thy prophets, and digged down thine altars; and I am left alone, and they seek my life. But what saith the answer of God unto him? I have reserved to myself seven thousand men, who have not bowed the knee to the image of Baal. Even so then at this present time also there is a remnant according to the election of grace. And if by grace, then is it no more of works: otherwise grace is no more grace. But if it be of works, then is it no more grace: otherwise work is no more work. What then? Israel hath not obtained that which he seeketh for; but the election hath obtained it, and the rest were blinded (According as it is written, God hath given them the spirit of slumber, eyes that they should not see, and ears that they should not hear;) unto this day. And David saith, Let their table be made a snare, and a trap, and a stumblingblock, and a recompence unto them: Let their eyes be darkened, that they may not see, and bow down their back alway. I say then, Have they stumbled that they should fall? God forbid: but rather through their fall salvation is come unto the Gentiles, for to provoke them to jealousy. Now if the fall of them be the riches of the world, and the diminishing of them the riches of the Gentiles; how much more their fulness? For I speak to you Gentiles, inasmuch as I am the apostle of the Gentiles, I magnify mine office: If by any means I may provoke to emulation them which are my flesh, and might save some of them. For if the casting away of them be the reconciling of the world, what shall the receiving of them be, but life from the dead? For if the firstfruit be holy, the lump is also holy: and if the root be holy, so are the branches. And if some of the branches be broken off, and thou, being a wild olive tree, wert grafted in among them, and with them partakest of the root and fatness of the olive tree; Boast not against the branches. But if thou boast, thou bearest not the root, but the root thee. Thou wilt say then, The branches were broken off, that I might be grafted in. Well; because of unbelief they were broken off, and thou standest by faith. Be not highminded, but fear: For if God spared not the natural branches, take heed lest he also spare not thee. Behold therefore the goodness and severity of God: on them which fell, severity; but toward thee, goodness, if thou continue in his goodness: otherwise thou also shalt be cut off. And they also, if they abide not still in unbelief, shall be grafted in: for God is able to graft them in again. For if thou wert cut out of the olive tree which is wild by nature, and wert grafted contrary to nature into a good olive tree: how much more shall these, which be the natural branches, be grafted into their own olive tree? (Romans 11:1-24)
Paul taught that it was the root (the covenant connection to God through his word) that empowered the blessings of the people, not the branch (the people themselves). Paul's discourse conveys the same ideas as Jacob 5, which describes the constant process of tying in people to covenant blessings, and removing people from those blessings because of their failure to engender the works of righteousness despite the opportunity for a connection with heaven.
Behold, I say unto you, that the house of Israel was compared unto an olive tree, by the Spirit of the Lord which was in our father; and behold are we not broken off from the house of Israel, and are we not a branch of the house of Israel? And now, the thing which our father meaneth concerning the grafting in of the natural branches through the fulness of the Gentiles, is, that in the latter days, when our seed shall have dwindled in unbelief, yea, for the space of many years, and many generations after the Messiah shall be manifested in body unto the children of men, then shall the fulness of the gospel of the Messiah come unto the Gentiles, and from the Gentiles unto the remnant of our seed-And at that day shall the remnant of our seed know that they are of the house of Israel, and that they are the covenant people of the Lord; and then shall they know and come to the knowledge of their forefathers, and also to the knowledge of the gospel of their Redeemer, which was ministered unto their fathers by him; wherefore, they shall come to the knowledge of their Redeemer and the very points of his doctrine, that they may know how to come unto him and be saved. And then at that day will they not rejoice and give praise unto their everlasting God, their rock and their salvation? Yea, at that day, will they not receive the strength and nourishment from the true vine? Yea, will they not come unto the true fold of God? Behold, I say unto you, Yea; they shall be remembered again among the house of Israel; they shall be grafted in, being a natural branch of the olive tree, into the true olive tree. And this is what our father meaneth; and he meaneth that it will not come to pass until after they are scattered by the Gentiles; and he meaneth that it shall come by way of the Gentiles, that the Lord may show his power unto the Gentiles, for the very cause that he shall be rejected of the Jews, or of the house of Israel. (1 Nephi 15:12-17)
In our day, we choose whether we will accept the words of God, and allow ourselves to thus be grafted into the true vine, and bear choice fruit, or whether we will cut ourselves off from the true vine by stifling or refusing God's word.