The New Covenant in Jeremiah 31

The New Testament, or the New Covenant, first appears in Jeremiah 31. The author of Hebrews quoted extensively from Jeremiah to prove that the previous covenant, the Old Covenant, has been rendered obsolete by the establishment of the New Covenant.
But as it is, Christ has obtained a ministry which is as much more excellent than the old as the covenant he mediates is better, since it is enacted on better promises.
For if that first covenant had been faultless, there would have been no occasion for a second (Hebrews 8:6-7).
The author found the Archille's heel of the Old Covenant from Jeremiah, the unpopular prophet. He quoted in length Jeremiah 31:31-34 which articulates what the author of Hebrews wanted to say. The house of Israel and the house of Judah (i.e. the Divided Kingdom) did not continue in God's covenant. So, God paid no heed to them and did not hear their prayers, nor accept their offerings. God had abandoned their Temple cult.
The above is the Christian interpretation of Jeremiah 31. What about the Jewish Rabbis? How should they handle these Christian prooftexts, such as Psalm 95, Psalm 110 and Jeremiah 31? I hope someday, someone will enlighten me on how the Jewish Rabbis interpret these texts which Christians have borrowed to support their Christian views and even to attack Judaism. Indeed, they are respectable scholars. Instead of removing the offending chapters, they allow them to stay and work hard on a better interpretation instead.
Looking back into history, Martin Luther had chosen a very damaging tactic. He was heroic in challenging the corruption in the Middle Age Catholic Church and in condemning the sale of indulgence. As a theologian, he wanted to remove its roots as well, i.e. the theology of purgatory, of praying for the dead etc. These ideas can be found in the books of Macabees. What is more, only Greek translations of these books exist. The Hebrew originals were lost. So, Martin Luther proposed to remove Old Testament books that did not have the Hebrew original. Consequently, 7 books were axed from the Old Testament of the Protestant Bible. This was indeed a damaging act. Luther had thrown the baby out with the bath water! These Greek texts are in fact a bridge between the two Testaments. In his enthusiasm to clean the teachings of the Church, Luther had created many missing links for the Protestant tradition.

Thank you Lord for inviting us to join the New Covenant. Write Your law of charity into our hearts so that we can love You and our neighbours more. Amen.

Appendix:
The days will come, says the Lord, when I will establish a new covenant with the house of Israel and with the house of Judah;
not like the covenant that I made with their fathers on the day when I took them by the hand to lead them out of the land of Egypt; for they did not continue in my covenant, and so I paid no heed to them, says the Lord.
This is the covenant that I will make with the house of Israel after those days, says the Lord: I will put my laws into their minds, and write them on their hearts, and I will be their God, and they shall be my people.
And they shall not teach every one his fellow or every one his brother, saying, `Know the Lord, for all shall know me, from the least of them to the greatest.
For I will be merciful toward their iniquities, and I will remember their sins no more.
ἰδοὺ ἡμέραι ἔρχονται, λέγει κύριος,καὶ συντελέσω ἐπὶ τὸν οἶκον Ἰσραὴλ καὶ ἐπὶ τὸν οἶκον Ἰούδα διαθήκην καινήν,
οὐ κατὰ τὴν διαθήκην, ἣν ἐποίησα τοῖς πατράσιν αὐτῶν ἐν ἡμέρᾳ ἐπιλαβομένου μου τῆς χειρὸς αὐτῶν ἐξαγαγεῖν αὐτοὺς ἐκ γῆς Αἰγύπτου,ὅτι αὐτοὶ οὐκ ἐνέμειναν ἐν τῇ διαθήκῃ μου, κἀγὼ ἠμέλησα αὐτῶν, λέγει κύριος·
ὅτι αὕτη ἡ διαθήκη, ἣν διαθήσομαι τῷ οἴκῳ Ἰσραὴλ μετὰ τὰς ἡμέρας ἐκείνας, λέγει κύριος· διδοὺς νόμους μου εἰς τὴν διάνοιαν αὐτῶν καὶ ἐπὶ καρδίας αὐτῶν ἐπιγράψω αὐτούς, καὶ ἔσομαι αὐτοῖς εἰς θεόν, καὶ αὐτοὶ ἔσονταί μοι εἰς λαόν·
καὶ οὐ μὴ διδάξωσιν ἕκαστος τὸν πολίτην αὐτοῦ καὶ ἕκαστος τὸν ἀδελφὸν αὐτοῦ λέγων· γνῶθι τὸν κύριον, ὅτι πάντες εἰδήσουσίν με ἀπὸ μικροῦ ἕως μεγάλου αὐτῶν,
ὅτι ἵλεως ἔσομαι ταῖς ἀδικίαις αὐτῶν καὶ τῶν ἁμαρτιῶν αὐτῶν οὐ μὴ μνησθῶ ἔτι.
(Hebrews 8:8b-12)
Ἰδοὺ ἡμέραι ἔρχονται, φησὶν κύριος, καὶ διαθήσομαι τῷ οἴκῳ Ισραηλ καὶ τῷ οἴκῳ Ιουδα διαθήκην καινήν,
οὐ κατὰ τὴν διαθήκην, ἣν διεθέμην τοῖς πατράσιν αὐτῶν ἐν ἡμέρᾳ ἐπιλαβομένου μου τῆς χειρὸς αὐτῶν ἐξαγαγεῖν αὐτοὺς ἐκ γῆς Αἰγύπτου, ὅτι αὐτοὶ οὐκ ἐνέμειναν ἐν τῇ διαθήκῃ μου, καὶ ἐγὼ ἠμέλησα αὐτῶν, φησὶν κύριος,
ὅτι αὕτη ἡ διαθήκη, ἣν διαθήσομαι τῷ οἴκῳ Ισραηλ μετὰ τὰς ἡμέρας ἐκείνας, φησὶν κύριος Διδοὺς δώσω νόμους μου εἰς τὴν διάνοιαν αὐτῶν καὶ ἐπὶ καρδίας αὐτῶν γράψω αὐτούς, καὶ ἔσομαι αὐτοῖς εἰς θεόν, καὶ αὐτοὶ ἔσονταί μοι εἰς λαόν,
καὶ οὐ μὴ διδάξωσιν ἕκαστος τὸν πολίτην αὐτοῦ καὶ ἕκαστος τὸν ἀδελφὸν αὐτοῦ λέγων Γνῶθι τὸν κύριον, ὅτι πάντες εἰδήσουσίν με ἀπὸ μικροῦ αὐτῶν καὶ ἕως μεγάλου αὐτῶν, ὅτι ἵλεως ἔσομαι ταῖς ἀδικίαις αὐτῶν καὶ τῶν ἁμαρτιῶν αὐτῶν οὐ μὴ μνησθῶ ἔτι. (Jeremiah 38:31-34 LXX, 31:31-34 MT).