Why does Melchizedek represent something that is far superior to what was offered to ancient Israel under the Old Covenant? How is Jesus' role as mankind's mediator and intercessor in heaven critically linked to Melchizedek?
The only physical appearance of the man, from whom the order of Melchizedek is named, takes place in the lifetime of Abram (Abraham). In his early eighties (see Genesis 12:4), Abram is forced to muster up a small army of his trained servants. He does this to rescue Lot from a four-king alliance that has taken him and many others as prisoners (Genesis 14:1 - 14).
Abram is ultimately victorious in battle. He then meets a mysterious and highly honored priest who blesses him.
And Melchizedek the King of Salem brought forth bread and wine. And He was the Priest of the Most High God. And He blessed him, and said, 'Blessed be Abram of the Most High God . . . And he (Abram) gave Him tithes of all (Genesis 14:18 - 20, HBFV).
What can we learn about Melchizedek from this brief account? His name, in Hebrew, means someone who is the King of Right or Righteousness (Strong's Concordance #H4442), a fact later confirmed (Hebrews 7:2). He possesses the title "King of Salem" which, given that Salem means "peace" (Brown-Driver-Briggs Hebrew Definitions, #H8004), is synonymous with "King of Peace" (Hebrews 7:2). Salem is also an early name for the city of Jerusalem.
Paul's hidden proof
The Apostle Paul quotes a somewhat hidden Old Testament passage to show that the priesthood started by Melchizedek made possible Jesus' role as our heavenly high priest. It is a role that he did not perfectly fulfill until after he was born, suffered, died and was resurrected from the dead (Hebrews 5:7 - 10, 9:11 - 12).
So Christ also did not take upon himself the glory of becoming a high priest. But God said to him, "You are my Son; today I have become your Father. And he says in another place (Psalm 110:4), "You are a priest forever, in the order of Melchizedek" (Hebrews 5:5 - 6, NIV).
God's law specified that only those born of the tribe of Levi could serve as his priests (Numbers 8). Of these Levites, only those who were descended from Aaron were eligible to become High Priest (Exodus 29:9, 29 - 30, Leviticus 8:12, 16:32).
Jesus, however, came from the tribe of Judah, a tribe which Scripture says nothing concerning the priesthood (Hebrews 7:13 - 14). He was not eligible to be a priest. How could he then "legally" serve in such a capacity after his resurrection? The answer is the creation of the order of Melchizedek.
Why did God do it?
God intended, in advance, that the Old Covenant Levitical priesthood last for only a short time (Hebrews 7:11 - 12, 9 - 10) and be replaced. This is why the Biblical appearance of Melchizedek (Genesis 14:18 - 20) occurred many decades before Levi (Abraham's great-grandson) was born and more than 300 years prior to Israel receiving the law (Exodus 20).
The existence of his order, prior to the giving of the law, meant that it would not be bound by its rules regarding the priesthood. This made it possible for Jesus to serve, after his resurrection, as High Priest before God.
The Apostle Paul delineates some of the profound ways this order, begun by Melchizedek, is superior to what was instituted under the Old Covenant (Hebrews 7:7). It is a royal or ruling High priesthood (Hebrews 7:1), which is perfect (verse 11), changeless (verse 24) and composed of one individual who always existed (verse 3) and always will (verses 8, 16, 24).
Instead of intercession being accomplished periodically within an earthly temple (see Hebrews 9), it is performed constantly next to God's throne (Hebrews 7:25 - 26, 8:1 - 2, 10:11 - 12). The covenant that is mediated by the order of Melchizedek is much better (Hebrews 8:6), with better promises, than what was offering under the Old Covenant.