The Church of the NT fulfills the promises made to Israel because the promises center in Christ the Head of His Body the Church, and He is Israel. Notice some scripture that identifies Jesus Christ the Head of His Body the Church as Israel Hosea 11:1
When Israel was a child, then I loved him, and called my son out of Egypt.
Now lets read the fulfillment Matt 2:13-15
13 And when they were departed, behold, the angel of the Lord appeareth to Joseph in a dream, saying, Arise, and take the young child and his mother, and flee into Egypt, and be thou there until I bring thee word: for Herod will seek the young child to destroy him.
14 When he arose, he took the young child and his mother by night, and departed into Egypt:
15 And was there until the death of Herod: that it might be fulfilled which was spoken of the Lord by the prophet, saying, Out of Egypt have I called my son.
Now it should be plain to any reasonable person Jesus Christ is being identified here as the fulfillment of a prophecy about Israel !
Now Christ and His Body the Church are One, Gods word intimates. How can the Head be Identified as Israel and His Body not ? For Christ is the True Israel of God with His Members Chosen in Him. It is He that Israel that Glorified God as Gods servant Isa 49:1-3
Listen, O isles, unto me; and hearken, ye people, from far; The Lord hath called me from the womb; from the bowels of my mother hath he made mention of my name.
2 And he hath made my mouth like a sharp sword; in the shadow of his hand hath he hid me, and made me a polished shaft; in his quiver hath he hid me;
3 And said unto me, Thou art my servant, O Israel, in whom I will be glorified.
Did not Jesus Christ become a servant ? Phil 2:5-8
5 Let this mind be in you, which was also in Christ Jesus:
6 Who, being in the form of God, thought it not robbery to be equal with God:
7 But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men:
8 And being found in fashion as a man, he humbled himself, and became obedient unto death, even the death of the cross.
Did not Jesus Glorify God the Father in the earth ? Jn 17:1-4
These words spake Jesus, and lifted up his eyes to heaven, and said, Father, the hour is come; glorify thy Son, that thy Son also may glorify thee:
2 As thou hast given him power over all flesh, that he should give eternal life to as many as thou hast given him.
3 And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.
4 I have glorified thee on the earth: I have finished the work which thou gavest me to do.
Israel means in the hebrew "God prevails" In Christ God prevails
Israel means in the greek "he shall be a prince of God" Christ is Prince Acts 5:31
31 Him hath God exalted with his right hand to be a Prince and a Saviour, for to give repentance to Israel, and forgiveness of sins.
Israel here is His Body the Church, because He is the Saviour of His Body Eph 5:23
For the husband is the head of the wife, even as Christ is the head/prince of the church: and he is the saviour of the body.