These teachings are due to a misunderstanding of the identity of the "True Israel of God."
The identity of the “Israel of God,” begins by realizing that whenever "Jacob" is mentioned in Scripture, he is always referred to by his natural birth name, “Jacob,” and not by the heavenly invoked name, “Israel.” The only time “Israel” is used is when the Bible is speaking of the 'spiritual Israel of God.' Jesus and Jesus alone lived up to the definition of the name "Israel," which is: “HE WILL RULE AS GOD” (See Strong's H3478). Jacob never lived up to that name nor did any of his descendants. Notice in the next scriptures how they all equate out to the same opinion of Jesus being the True Israel of God:
Matthew 2:14-15
(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.
Hosea 11:1
(1) WHEN ISRAEL WAS A CHILD, then I loved him, and CALLED MY SON OUT OF EGYPT.
Exodus 4:22
(22) And thou shalt say unto Pharaoh, Thus saith the LORD, ISRAEL IS MY SON, EVEN MY FIRSTBORN:
John 3:16
(16) For God so loved the world, that HE GAVE HIS ONLY BEGOTTEN SON, that whosoever believeth in him should not perish, but have everlasting life.
Mark 12:6-9
(6) Having yet therefore ONE SON, his wellbeloved, he sent him also last unto them, saying, They will reverence my son.
(7) But those husbandmen said among themselves, THIS IS THE HEIR; come, let us kill him, and the inheritance shall be ours.
(8 ) And they took him, and killed him, and cast him out of the vineyard.
(9) What shall therefore the lord of the vineyard do? HE WILL COME AND DESTROY THE HUSBANDMEN, AND WILL GIVE THE VINEYARD UNTO OTHERS.
These verses interrelate together showing that the Promised Child was and is Jesus Christ! He is the same promise Paul referred to allegorically in
Galatians 3:16 as the One promised SEED of Abraham. Through Jesus, we become partakers of that same promise, and members of the True Israel of God.
Neither the ethnicity of the Jews nor their opportunity to receive salvation through Jesus ended because of the 70 AD destruction. But with this destruction came the elimination of the self-righteous system of Law keeping to which the Jews had fallen prey. Jesus' coming in judgment during that generation – as prophesied – firmly established Him as the King of kings and the Lord of lords who does RULE AS GOD!
An identity crisis comes from a lack of identity in Chirst. Once a person sees that Jesus is the True Israel of God, they can then be set free to pursue Christlikeness. That is what Paul meant by this statement:
Galatians 6:15-16
(15) For in Christ Jesus neither circumcision availeth any thing, nor uncircumcision, but a new creature.
(16) And as many as walk according to this rule, peace be on them, and mercy, and upon the Israel of God.
It’s all about being IN JESUS CHRIST! The "rule" to which verse 16 speaks is not about being a Jew (circumcision) or Gentile (uncircumcision), but about being a "New Creature" in Christ! This is speaking of New Covenant, not Old Covenant. Paul is saying that those who walk in THAT rule will have "peace," "mercy." The conjunction "and" tying this to Paul’s "Israel of God," is the Greek work “Kai.” Kai can be translated as "and, also, even, indeed, but" (See G2532). The Amplified Bible uses the word “even” in place of “and” in this verse. This shows Paul is saying that those living by this rule make up those who are the Israel of God.