Of the five covenants made between God and man (Adamic, Noahic, Abrahamic, Mosaic, Davidic), none were intended to procure fellowship with Him, but were used to prepare and lead men to the one covenant which would procure fellowship with Him, which is the Everlasting Covenant (Heb 13:20). This is the “New Testament or Covenant, as the modern translations read, that Jesus spoke of in His blood (Mat 26:28; Mark 14:24; Luke 22:20). Nowhere is this Covenant shown to be between us and Christ or us and the Father but rather, we are the recipients of this Covenant.
This is the only Covenant made that was not between God and man because it was made in eternity past—between the Father and the Son. This involved the Father’s promise to raise Jesus from the dead after atoning for the sins of man; “the covenant that was confirmed before by God in Christ” (Gal 3:17). “And for this reason He is the Mediator of the new covenant, by means of death, for the redemption of the transgressions under the first covenant, that those who are called may receive the promise of the eternal inheritance” (Heb 9:15).
The purpose of God's dealings with Israel was to reveal to them and the rest of the world His desire and plans for fellowship with Him. Jesus’ atonement for sin broke the curse incurred by Adam and Eve (Gen 2:17), which means not only Israel’s redemption “from the curse of the law” (most have yet to attain) but also “That the blessing of Abraham might come on the Gentiles through Jesus Christ” for “the scripture hath concluded all under sin, that the promise by faith of Jesus Christ might be given to them that believe”. (Gal 3:13, 14, 22).
“Now may the God of peace who brought up our Lord Jesus from the dead, that great Shepherd of the sheep, through the blood of the everlasting covenant, make you complete in every good work to do His will, working in you what is well pleasing in His sight, through Jesus Christ, to whom be glory forever and ever. Amen (Heb 13:20, 21).