Shared ground
Hebrews 9:15–17 links the new covenant directly to Jesus’ death. The text states that Jesus is the mediator of the new covenant, that a death has taken place, and that this death is connected to “redemption” for wrongs committed under the first covenant (v.15). The stated outcome is that “those who have been called” may receive the promised “eternal inheritance” (v.15).
The passage then gives a supporting explanation from common experience: a will only becomes effective when the one who made it dies (vv.16–17). Death is presented as the condition that brings the promised transfer into effect.
Where interpretation differs (only where needed)
1) What diathēkē means in vv.16–17. Some readers think the word keeps the meaning “covenant” throughout, and vv.16–17 are an illustration drawn from how covenants are enacted with death. Others think the author intentionally shifts to the everyday sense “will/testament” in vv.16–17 to make a clear point about inheritance.
2) How the “will” illustration connects back to v.15. Some take vv.16–17 as directly explaining why covenant benefits require the mediator’s death. Others see vv.16–17 mainly explaining the “inheritance” language in v.15: inheritances are received when the one whose estate is given has died, so the promised inheritance comes through Jesus’ death.
3) What “redemption of transgressions under the first covenant” includes. Some read it narrowly: it addresses the accumulated guilt of people living under the first covenant and shows that Christ’s death reaches back to cover those sins. Others read it more broadly: it means the new covenant resolves the problem that the first covenant exposed but could not finally remove.
Why the disagreement exists
The disagreement mostly comes from one word: diathēkē can be used for “covenant,” and it can also be used for something like a “will/testament.” Hebrews is already talking about covenants in the surrounding context, but vv.16–17 use reasoning that fits the ordinary practice of wills in the Greco-Roman world. Interpreters differ on whether the author is keeping a single meaning or using a deliberate wordplay to connect covenant and inheritance.
What this passage clearly contributes
Explicitly, the passage claims that Jesus’ death is not an accident alongside the new covenant; it is necessary for the promised benefits to be received (vv.15–17). It also clarifies what those benefits include: redemption related to sins committed under the first covenant, and the promised eternal inheritance for the called (v.15).
By inference, the logic suggests that the new covenant is established and its promised “inheritance” is released through the mediator’s death, in a way comparable to how a will only takes effect when the maker dies (vv.16–17).