The June issue of Ligonier Ministries’ Tabletalk magazine features a daily study through Hebrews, starting with ch. 8 on June 1. I don’t know who wrote the study, but they make some very good points, quoting Owen several times.
8:4 is saying that because Christ is a member of the superior priesthood, His work cannot be done on earth. To engage in priestly ministry on earth is the province of the inferior Levitical priesthood, which is exercised according to the Mosaic law, the law that cannot perfect anyone.
John Owen says these better promises are the new covenant promises of Jeremiah 31:33–34 that God will write His law on the hearts of His people and remember their sin no more, that is, finally and fully forgive them. These promises were not fulfilled by the old covenant mediated by Moses. After all, the repeated sacrifices of the Mosaic law mean that the old covenant could not provide full and final forgiveness. They, and the old covenant of which they were a part, could only remind people of sin, not remove it (Heb. 10:1–18). Furthermore, the law demonstrates that the old covenant cannot be the means by which God writes His commandments on the hearts of His people. Deuteronomy 31:14–29 foresees that Israel as a nation would be so corrupt as to break the old covenant. The people would need a new heart, a heart that would come only after the nation of Israel broke the old covenant and suffered the curse of exile (30:1–10).
Nevertheless, the reality of the new covenant promises belonged to the old covenant saints. After all, David, an old covenant believer, enjoyed the complete and final forgiveness of sins in his justification (Rom. 4:5–8). No one is saved except through Christ and His new covenant, which is the ultimate expression of the one covenant of grace between God and His people (John 14:6). The old covenant saints belonged also to the one covenant of grace, though they lived prior to the inauguration of the new covenant. They, no less than us, were redeemed by Jesus alone, though their understanding of this was less full than is ours as new covenant believers.
The new covenant is necessary, Hebrews 8:8 tells us, because God found fault with the people. Our Creator never intended the old covenant to bring the blessings we have under the new covenant, though the old covenant saints possessed the benefits of the new covenant, albeit to a lesser degree than we do…
In and through the new covenant, we get what the subcovenants of the covenant of grace hoped for. The new covenant, Hebrews 8:10 reveals, is the means by which God’s promise to be God to Abraham is accomplished (see Gen. 17:7).
While disagreeing with the author’s interpretation of Gen 17:7 (see here), he correctly notes that Abraham was saved by the new covenant. Abraham, and all other Old Testament saints, received the promises/blessings of the new covenant in advance of its formal establishment.