Why is Jeremiah 31:31-34 interpreted by Catholic Christians to point toward Jesus?

Question

Why is Jeremiah 31:31-34 interpreted by Catholic Christians to point toward Jesus? How and why has Christianity fused together the idea of a Hebrew מָשִׁיחַ, or messiah, with the concept of something that God will do? Is the verse really pointing to a prophecy that will be fulfilled by a מָשִׁיחַ, or are Christians just vulnerable to the hindsight bias?

Please give the answer in English!

Not like the covenant that I formed with their forefathers on the day I took them by the hand to take them out of the land of Egypt, that they broke My covenant, although I was a lord over them, says the Lord.

For this is the covenant that I will form with the house of Israel after those days, says the Lord: I will place My law in their midst and I will inscribe it upon their hearts, and I will be their God and they shall be My people.

And no longer shall one teach his neighbor or [shall] one [teach] his brother, saying, "Know the Lord," for they shall all know Me from their smallest to their greatest, says the Lord, for I will forgive their iniquity and their sin I will no longer remember.

לֹא כַבְּרִית אֲשֶׁר כָּרַתִּי אֶת אֲבוֹתָם בְּיוֹם הֶחֱזִיקִי בְיָדָם לְהוֹצִיאָם מֵאֶרֶץ מִצְרָיִם אֲשֶׁר הֵמָּה הֵפֵרוּ אֶת בְּרִיתִי וְאָנֹכִי בָּעַלְתִּי בָם נְאֻם יְהֹוָה:

כִּי זֹאת הַבְּרִית אֲשֶׁר אֶכְרֹת אֶת בֵּית יִשְׂרָאֵל אַחֲרֵי הַיָּמִים הָהֵם נְאֻם יְהֹוָה נָתַתִּי אֶת תּוֹרָתִי בְּקִרְבָּם וְעַל לִבָּם אֶכְתֳּבֶנָּה וְהָיִיתִי לָהֶם לֵאלֹהִים וְהֵמָּה יִהְיוּ לִי לְעָם:

וְלֹא יְלַמְּדוּ עוֹד אִישׁ אֶת רֵעֵהוּ וְאִישׁ אֶת אָחִיו לֵאמֹר דְּעוּ אֶת יְהֹוָה כִּי כוּלָּם יֵדְעוּ אוֹתִי לְמִקְּטַנָּם וְעַד גְּדוֹלָם נְאֻם יְהֹוָה כִּי אֶסְלַח לַעֲו‍ֹנָם וּלְחַטָּאתָם לֹא אֶזְכָּר עוֹד:

-- Jeremiah 31:31-33, chabad.org

Answer 1

The verses of Jeremiah 31:31–34 appear in seven paragraphs in the Catechism of the Catholic Church, of which six (quoted below) are directly relevant. In all of these, Christ the Messiah is the bringer of the New Covenant, which is to be written on the hearts of his people.

This is entirely Biblical. The writer of the Epistle to the Hebrews quotes the passage verbatim in chapter 8, and then in chapter 9 describes the differences between the High Priest of the Old Covenant and the High Priest of the New, concluding:

Therefore he is the mediator of a new covenant, so that those who are called may receive the promised eternal inheritance, since a death has occurred which redeems them from the transgressions under the first covenant. [Heb 9:15 RSVCE]

Not only is the Jeremiah prophecy pointing to a prophecy which will be fulfilled by a Messiah, it points to the Messiah fulfilling the prophecy by providing the New Covenant.

64 Through the prophets, God forms his people in the hope of salvation, in the expectation of a new and everlasting Covenant intended for all, to be written on their hearts.

368 The spiritual tradition of the Church also emphasizes the heart, in the biblical sense of the depths of one's being, where the person decides for or against God.

580 In Jesus, the Law no longer appears engraved on tables of stone but "upon the heart" of the Servant who becomes "a covenant to the people", because he will "faithfully bring forth justice".

715 The prophetic texts that directly concern the sending of the Holy Spirit are oracles by which God speaks to the heart of his people in the language of the promise, with the accents of "love and fidelity." St. Peter will proclaim their fulfillment on the morning of Pentecost. According to these promises, at the "end time" the Lord's Spirit will renew the hearts of men, engraving a new law in them. He will gather and reconcile the scattered and divided peoples; he will transform the first creation, and God will dwell there with men in peace.

762 But the prophets accuse Israel of breaking the covenant and behaving like a prostitute. They announce a new and eternal covenant. "Christ instituted this New Covenant."

1965 The New Law or the Law of the Gospel is the perfection here on earth of the divine law, natural and revealed. It is the work of Christ and is expressed particularly in the Sermon on the Mount. It is also the work of the Holy Spirit and through him it becomes the interior law of charity: "I will establish a New Covenant with the house of Israel. . . . I will put my laws into their hands, and write them on their hearts, and I will be their God, and they shall be my people."

[The seventh is paragraph 2713, about contemplative prayer.]