Why should a Jew be interested in the New Covenant / Brit Chadasha? 

Quoting the nabi / prophet Yirmiyahu from the Tanakh:

30Behold, days are coming, says the Lord, and I will form a covenant with the house of Israel and with the house of Judah, a new covenant.		להִנֵּה יָמִים בָּאִים נְאֻם יְהֹוָה וְכָרַתִּי אֶת בֵּית יִשְׂרָאֵל וְאֶת בֵּית יְהוּדָה בְּרִית חֲדָשָׁה:
31Not 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.		לאלֹא כַבְּרִית אֲשֶׁר כָּרַתִּי אֶת אֲבוֹתָם בְּיוֹם הֶחֱזִיקִי בְיָדָם לְהוֹצִיאָם מֵאֶרֶץ מִצְרָיִם אֲשֶׁר הֵמָּה הֵפֵרוּ אֶת בְּרִיתִי וְאָנֹכִי בָּעַלְתִּי בָם נְאֻם יְהֹוָה:
32For 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.		לבכִּי 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:
33And 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.		לגוְלֹא יְלַמְּדוּ עוֹד אִישׁ אֶת רֵעֵהוּ וְאִישׁ אֶת אָחִיו לֵאמֹר דְּעוּ אֶת יְהֹוָה כִּי כוּלָּם יֵדְעוּ אוֹתִי לְמִקְּטַנָּם וְעַד גְּדוֹלָם נְאֻם יְהֹוָה כִּי אֶסְלַח

This is not to say, throw out your brains, throw away your Jewishness and go join the first Christian Church you encounter. For just as the teachers (rabbiym) of the Old Covenant have erred greatly, so have the goyum run teaching of the New Covenant... beginning with the fact that G-d intended the New Covenant to be as Jewish as the Old Covenant. That Israel was not to join the Church, but the Church was to join Israel (Spirit Israel). 

Read the text again. The New Covenant is with the House of Israel, House of Judah (verse 30), House of Israel (verse 32).  

In the Brit Chadasha Rabbi Shaul wrote:

Romans 11:16–27 (AV)

16 For if the firstfruit be holy, the lump is also holy: and if the root be holy, so are the branches. 

 

17 And if some of the branches be broken off, and thou, being a wild olive tree, wert graffed in among them, and with them partakest of the root and fatness of the olive tree; 

18 Boast not against the branches. But if thou boast, thou bearest not the root, but the root thee. 

19 Thou wilt say then, The branches were broken off, that I might be graffed in. 

20 Well; because of unbelief they were broken off, and thou standest by faith. Be not highminded, but fear: 

21 For if God spared not the natural branches, take heed lest he also spare not thee. 

22 Behold therefore the goodness and severity of God: on them which fell, severity; but toward thee, goodness, if thou continue in his goodness: otherwise thou also shalt be cut off. 

23 And they also, if they abide not still in unbelief, shall be graffed in: for God is able to graff them in again. 

24 For if thou wert cut out of the olive tree which is wild by nature, and wert graffed contrary to nature into a good olive tree: how much more shall these, which be the natural branches, be graffed into their own olive tree? 

25 For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in. 

26 And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob: 

27 For this is my covenant unto them, when I shall take away their sins.  

This is not replacement theology. Just the opposite. It is inclusion theology. That all who believe in the Jewish Messiah (as promised by Moshe in Devarim 18:15-19) will be saved into the House of Israel (Yirmiyahu 31:30-33). 

A form of replacement theology is what the Church has been suffering from. But rather all who believe in Messiah (Jew or Gentile) become Spirit Jews of the Spirit House of Israel. 

This does not make Gentiles into physical Jews. One has to be born a physical Jew. And it does not entitle Gentile believers (Spirit Jews) to any of the land grants of G-d in Eretz whatsoever. Nor does faith in the Jewish Messiah detract from the Jewishness one was born with.  

Speaking of being a Spirit Jew, Rabbi Shaul also wrote:

Romans 2:28–29 (AV)

28 For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh:

29 But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God.