The new covenant had to do with a deeper INCARNATION of God into people. Hebrews (8,9) is describing the nature of the new covenant in terms of greater (ever increasing!) incarnation. For, unlike the old covenant, in this new one, God directly puts His Law (Mind) inside people, and writes it upon their hearts, and God is, eventually, known directly in our midst. It is as if Jesus initiated an infusion of God into His Creation! He paid the price to open the Gate (which He is and was!) for the world of The Father to enter ALL of His Creation! The new covenant is then about a much deeper communion God desires with His creation. And it is also the Way into this deeper knowing.
Paul places this in the context of a fading of the old covenant and the coming of the new covenant; this is the covenant which Jesus said He had come to in-act at the last supper in His Toast, where He quoted Jeremiah 31. Jesus is saying that, I am the new covenant, and it has begun-participate in this deeper communion through Me, for God has begun a work where He will know and be known fully by His Creation. My blood has sealed and started this work! Start to participate in it!
You have BECOME the temple space (as Paul puts it in I Cor 3:16). The bread in the sanctuary becomes the bread of Life; all the central metaphors become more incarnate in the new testimony (or covenant)—from physical/ literal—to—metaphysical/spiritual reality. Literal bread or manna, to “I am the bread of Life.” In the new covenant metaphor, you are seeing God’s pattern of shifting things towards greater and greater incarnation!
The new covenant is no longer God WITH us-nearby, up the mountain etc; law “over there”, on tablets etc; but GOD IN US—the law written in us; “My words have I placed in your hearts that you might not sin” etc; Here, is God living IN, and through us; through the life of Jesus He lives out His Life! God communing in us through His Son—that is the nature of the new covenant.
Jesus named this at the last supper. Quoting Jeremiah, He said that this is the new covenant sealed by My Blood! This type of communion was new and was about Him dwelling in us through His Spirit and communing directly into us-it was the beginning of a new type of covenant God was/is making with humanity!