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Why did Christ say He was not sent to anyone but the lost sheep of the house of Israel?

Why did Jesus claim to be sent for the salvation of the world then?

John 3:1-21 3 Now there was a Pharisee named Nicodemus, a leader of the Jews. 2 He came to Jesus[a] by night and said to him, “Rabbi, we know that you are a teacher who has come from God; for no one can do these signs that you do apart from the presence of God.” 3 Jesus answered him, “Very truly, I tell you, no one can see the kingdom of God without being born from above.”[b] 4 Nicodemus said to him, “How can anyone be born after having grown old? Can one enter a second time into the mother’s womb and be born?” 5 Jesus answered, “Very truly, I tell you, no one can enter the kingdom of God without being born of water and Spirit. 6 What is born of the flesh is flesh, and what is born of the Spirit is spirit.[c] 7 Do not be astonished that I said to you, ‘You[d] must be born from above.’[e] 8 The wind[f] blows where it chooses, and you hear the sound of it, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit.” 9 Nicodemus said to him, “How can these things be?” 10 Jesus answered him, “Are you a teacher of Israel, and yet you do not understand these things? 11 “Very truly, I tell you, we speak of what we know and testify to what we have seen; yet you[g] do not receive our testimony. 12 If I have told you about earthly things and you do not believe, how can you believe if I tell you about heavenly things? 13 No one has ascended into heaven except the one who descended from heaven, the Son of Man.[h] 14 And just as Moses lifted up the serpent in the wilderness, so must the Son of Man be lifted up, 15 that whoever believes in him may have eternal life.[i] 16 “For God so loved the world that he gave his only Son, so that everyone who believes in him may not perish but may have eternal life. 17 “Indeed, God did not send the Son into the world to condemn the world, but in order that the world might be saved through him. 18 Those who believe in him are not condemned; but those who do not believe are condemned already, because they have not believed in the name of the only Son of God. 19 And this is the judgment, that the light has come into the world, and people loved darkness rather than light because their deeds were evil. 20 For all who do evil hate the light and do not come to the light, so that their deeds may not be exposed. 21 But those who do what is true come to the light, so that it may be clearly seen that their deeds have been done in God.”[j]

When did the new covenant go into effect? And who was it made with?

The new covenant was made at the Passover before the Crucifixion and sealed by the very blood of God at the crucifixion; though God, like a skillful author, used foreshadowing.

Jeremiah 31:31-34 31 The days are surely coming, says the Lord, when I will make a new covenant with the house of Israel and the house of Judah. 32 It will not be like the covenant that I made with their ancestors when I took them by the hand to bring them out of the land of Egypt—a covenant that they broke, though I was their husband,[a] says the Lord. 33 But this is the covenant that I will make with the house of Israel after those days, says the Lord: I will put my law within them, and I will write it on their hearts; and I will be their God, and they shall be my people. 34 No longer shall they teach one another, or say to each other, “Know the Lord,” for they shall all know me, from the least of them to the greatest, says the Lord; for I will forgive their iniquity, and remember their sin no more.

and, like a master composer, He uses recapitulation:

Rev. 21:3-7 And I heard a loud voice from the throne saying, “See, the home[a] of God is among mortals. He will dwell[b] with them; they will be his peoples,[c] and God himself will be with them;[d] 4 he will wipe every tear from their eyes. Death will be no more; mourning and crying and pain will be no more, for the first things have passed away.” 5 And the one who was seated on the throne said, “See, I am making all things new.” Also he said, “Write this, for these words are trustworthy and true.” 6 Then he said to me, “It is done! I am the Alpha and the Omega, the beginning and the end. To the thirsty I will give water as a gift from the spring of the water of life. 7 Those who conquer will inherit these things, and I will be their God and they will be my children.

So, looking at the end of the book, the New Covenant is made with mortals, peoples, and as he said to Nicodemus the world.

Just reading Scripture will do no good unless it is compared Scripture with Scripture and rightly divided according to GOD’S COMMAND. 2 Tim. 2:15.

Funnily enough the verse right before says to avoid "lawyering" (because it turns off those who are interested) and, immediately after it, it says to avoid profane chatter which will spread like gangrene — heavily implying that you are taking that verse out of context:

14 Remind them of this, and warn them before God[a] that they are to avoid wrangling over words, which does no good but only ruins those who are listening. 15 Do your best to present yourself to God as one approved by him, a worker who has no need to be ashamed, rightly explaining the word of truth. 16 Avoid profane chatter, for it will lead people into more and more impiety, 17 and their talk will spread like gangrene. Among them are Hymenaeus and Philetus, 18 who have swerved from the truth by claiming that the resurrection has already taken place. They are upsetting the faith of some. 19 But God’s firm foundation stands, bearing this inscription: “The Lord knows those who are his,” and, “Let everyone who calls on the name of the Lord turn away from wickedness.”

The translation I'm using here, NRSV, says explaining [the word of] truth rather than dividing; which makes more sense in-context than 'dividing' (which could also be translated discerning or discriminating).

There is only one lawgiver, there is only one law; 'dividing' the word of God is not possible — 'dividing' the people of God is not good:

James 2:8-11; 4:1-12 8 You do well if you really fulfill the royal law according to the scripture, “You shall love your neighbor as yourself.” 9 But if you show partiality, you commit sin and are convicted by the law as transgressors. 10 For whoever keeps the whole law but fails in one point has become accountable for all of it. 11 For the one who said, “You shall not commit adultery,” also said, “You shall not murder.” Now if you do not commit adultery but if you murder, you have become a transgressor of the law. Those conflicts and disputes among you, where do they come from? Do they not come from your cravings that are at war within you? 2 You want something and do not have it; so you commit murder. And you covet[t] something and cannot obtain it; so you engage in disputes and conflicts. You do not have, because you do not ask. 3 You ask and do not receive, because you ask wrongly, in order to spend what you get on your pleasures. 4 Adulterers! Do you not know that friendship with the world is enmity with God? Therefore whoever wishes to be a friend of the world becomes an enemy of God. 5 Or do you suppose that it is for nothing that the scripture says, “God[u] yearns jealously for the spirit that he has made to dwell in us”? 6 But he gives all the more grace; therefore it says, “God opposes the proud, but gives grace to the humble.” 7 Submit yourselves therefore to God. Resist the devil, and he will flee from you. 8 Draw near to God, and he will draw near to you. Cleanse your hands, you sinners, and purify your hearts, you double-minded. 9 Lament and mourn and weep. Let your laughter be turned into mourning and your joy into dejection. 10 Humble yourselves before the Lord, and he will exalt you. 11 Do not speak evil against one another, brothers and sisters.[v] Whoever speaks evil against another or judges another, speaks evil against the law and judges the law; but if you judge the law, you are not a doer of the law but a judge. 12 There is one lawgiver and judge who is able to save and to destroy. So who, then, are you to judge your neighbor?