This is not a puzzle for theologians to keep among themselves. The gospel is news about a person, and who that person is decides what the news is worth.
Salvation comes through a mediator — “one God, and one mediator between God and men, the man Christ Jesus” (1 Timothy 2:5). A mediator who was not truly man could not stand in our place. A mediator who was not truly God could not bring us to the Father. Because He is both, Jesus can say, “I am the way, the truth, and the life: no man cometh unto the Father, but by me” (John 14:6), and the apostles can say there is “none other name under heaven given among men, whereby we must be saved” (Acts 4:12).
It also settles how you may come. Hebrews does not tell us to approach a distant figure cautiously. It tells us to come boldly — because the throne of grace is occupied by someone who has felt our infirmities and remained without sin. You are not explaining yourself to a stranger. You are trusting a High Priest who has been here.
And it makes the promise sure. His work is “once for all” and needs no supplement (Hebrews 10:10); He “ever liveth to make intercession” and so saves to the uttermost (Hebrews 7:25). The confidence of the believer never rests on the strength of his own grip. It rests on the identity of the one he is holding.
The response Scripture asks for is the one Thomas gave — to look at Jesus Christ and call Him “my Lord and my God” — and then to receive Him: “if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved” (Romans 10:9). “But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name” (John 1:12).