Did Jesus Himself make that claim? The only place in the NT which uses that phrase is in John's Gospel which places it in the mouth of the Samaritans, not Jesus. In any case, the world is clearly not saved.
Good question, Smaneck.
Matt 16:13-17: He asked Peter
"Whom do men say I, the son of man, am?", then
"Whom say ye that I am?" Following Peter's representative statement that Jesus was the Christ, the Savior said
"flesh and blood hath not revealed it unto thee, but my Father which is in heaven." (Mark 8:27-30; Luke 9:18-20; John 6:66-71)
Later he was emphatic in answering the chief priest's and Pilate's leading questions with direct, pointed, and unequivocal answer "Thou sayest it."
You say: a Jewish idiom of solemn affirmation, probably the origin of English idiom
“You said it!” meaning
that is very true. The idiom seems to call forth the eternal principle that
in the mouth of two or three witness shall every word be established. This phrase is not at all evasive. Matt 26:64-65; Mk 14:62-64; Lk 23:3; JST Mk 15:4
In fact, you missed the vital element in the Samaritan incident, John 4:25 -
26.
3 Nephi 11:10, 11
10 Behold, I am Jesus Christ, whom the prophets testified shall come into the world.
11 And behold, I am the light and the life of the world; and I have drunk out of that bitter cup which the Father hath given me, and have glorified the Father in taking upon me the sins of the world, in the which I have suffered the will of the Father in all things from the beginning.
Doctrine and Covenants 19:1, 16-19;
1 I am Alpha and Omega, Christ the Lord; yea, even I am he, the beginning and the end, the Redeemer of the world.
16 For behold, I, God, have suffered these things for all, that they might not suffer if they would repent;
17 But if they would not repent they must suffer even as I;
18 Which suffering caused myself, even God, the greatest of all, to tremble because of pain, and to bleed at every pore, and to suffer both body and spirit—and would that I might not drink the bitter cup, and shrink—
19 Nevertheless, glory be to the Father, and I partook and finished my preparations unto the children of men.