I'm sorry, but you're just using a religious definition of the name "Israel." In any dictionary or encyclopedia, or in the Bible itself, "Israel" refers to the nation that God called and placed in the land of Canaan. It doesn't matter who was faithful and who was not--all of those living in the land of Israel who were descended from the 12 tribes were "Israel."
Beyond that, unfaithful Israelis were to be cut off from their people. They didn't stop being Hebrews or "Israel," but they were cut off form the community, and thus were divorced from the nation.
Today Israel does not know their Messiah Jesus. But they are not being cut off because they were already dismissed in exile after 70 AD. They were "Israel in exile," waiting to be restored.
A remnant found Jesus as their Messiah, but the majority did not see Jesus as anything more than a religious figure whose followers persecuted them. They stubbornly resisted true testimonies that Jesus was their Messiah. And those who might have answered the call were discouraged from doing so by their own angry Jewish brethren.
But in God's time Israel will be restored to be God's people, this time among many other peoples. God promised Abraham many nations who would be called and chosen. Israel will be merely the last in a long string of them. Of course, minorities in non-Christian nations can be saved too, although like with Israel it is more difficult when you're pressured by your fellow citizens.
That above actually shows you have not studied all of your Bible, that you instead 'insert' of lot of men's theories into your understanding instead.
The new name "
Israel" which God gave to Jacob was in connection with God's Birthright that continued to Jacob. The Birthright was given first through Abraham, then through Isaac, and then to Jacob with it representing the PROMISE BY FAITH which Abraham believed, i.e, The Gospel of Jesus Christ, which God then counted Abraham's FAITH as RIGHTEOUSNESS. Apostle Paul specifically revealed this about Abraham in Galatians 3. Paul even said that those of FAITH have become the children of Abraham, because of the first FAITH by Abraham on the Promise by Faith which is about The Gospel.
Now orthodox Jews of course HATE this idea about Abraham's FAITH being the SAME Faith on Jesus Christ that we Christians have believed, but that is exactly what Apostle Paul taught in Galatians 3. The Promise by Faith of The Gospel was always FIRST, per God's written Word. This is why Lord Jesus said in John 8:56-58 that Abraham rejoiced to see Christ's day, and was glad. And per Jude 14-15 we are even shown that Enoch, the 7th from the man Adam, prophesied of Christ's still future 2nd coming with ten thousands of His saints to execute judgment upon the wicked. So don't try and tell me the Old Testament Patriarchs didn't know about Jesus and The Gospel. David in Psalms 22 and Isaiah in Isaiah 53 both were given to prophesy about Christ's crucifixion.
This is why Apostle Paul in Romans 9 revealed that God's TRUE Israel are those of FAITH. Just being born of the seed of the old historical nation of Israel is NOT ENOUGH. All... souls MUST ACCEPT JESUS OF NAZARETH AS GOD'S PROMISED SAVIOR to be saved.
God included Old Testament prophesy about the eventual manifesting of the Christian Church under the New Covenant. It especially involved the ten lost tribes of Israel that He scattered mainly to the West who would fulfill the "multitude of nations" Gen.48 prophecy about Ephraim. In Gen.35 God had previously told Jacob that He would make of Jacob's seed
"a nation, and a company of nations." That means ISRAELITE nations. That was fulfilled in Ephraim and the ten lost tribes of the
"house of Israel" by their accepting The Gospel of Jesus Christ in new lands. This prophecy has pieces of it also proclaimed first in the Book of Hosea which Apostle Paul even quoted from to believing Gentiles in Romans 9.
Even false Jews try to deny all this and twist what Jesus said. Jesus said He was not sent but unto the lost sheep of the "
house of Israel". That
"house of Israel" label after 1 Kings 11 became a label ONLY about the separated ten northern tribes of old Israel. The Jews represented ONLY the Israelites of the southern kingdom called the
"house of Judah". The majority of Jews rejected The Gospel, and thus the scattered ten tribe
"house of Israel" in new lands would accept... The Gospel, and form the western Christian nations of history, fulfilling the
"multitude of nations" Gen.48 prophecy to Ephraim.