Au contraire. Christ fulfilled the Old Covenant and instituted the New Covenant. In Christ, Israel consists of all believers.
"28 There is neither Jew nor Greek, there is neither slave[g] nor free, there is no male and female, for you are all one in Christ Jesus. 29 And if you are Christ's, then you are Abraham's offspring, heirs according to promise." Gal 3:28-29.
"For in Christ Jesus neither circumcision availeth any thing, nor uncircumcision, but a new creature. And whosoever shall follow this rule, peace on them, and mercy, and upon the Israel of God." - (Galatians 6:15-16)
The Israel of the OT prefigures the Israel of the NT. Mana kept the old Israelite's bodies alive, the bread of the NT, Christ's Flesh, keeps the new Israelite's souls alive. The old Israelites had a temple made with hands, the new Israelites in Christ have a temple made without hands, built in three days. The old Israelites were the 12 tribes of Israel, the new Israelites are all those, Jew and gentile, who are in Christ. The old Israelites were given the Jerusalem of the promises land, the new Israelites are given the heavenly Jerusalem of Christ's kingdom which is not of this earth. The old Israelites will one day be accept Christ en masse, and will be fully within the New Covenant, as are the new Israelites.
As a Christian, I am an Israelite of the New Covenant. I am of Zion, of the new Jerusalem.
And one note, "The Bible is...written by Israel"" is a statement which requires correction. The Bible is written by God Himself, not Israel.