Yes, this is true. There was no New Covenant during Jesus lifetime. Jesus is a Jew, and during his lifetime, He was under the Laws of Moses. Had Jesus taught any other Covenant other than the Mosaic Law, He would have been violating the Mosaic Covenant, and thereby did not teach what we Christians call the New Covenant. In fact, the Pharisees charged Jesus with violating the Mosaic Law, a charge which Jesus strongly defended himself against. Jesus ministry during His life was only to the Jews, and He showed the best approach for keeping all the commandments of Moses. The Sermon on the Mount was addressed to His fellow Jews. Yeshua addressed gentiles, telling them they are unworthy of eating the bread God set aside for His People, the offspring of Judah. He referred to gentiles as dogs unworthy to receive the bread God has for His people, the offspring of Judah. Yeshua did heal gentiles because when anyone comes to Him seeking mercy, God is merciful.
Matt 15:21-28
Jesus went away from there, and withdrew into the district of Tyre and Sidon. And a Canaanite woman from that region came out and began to cry out, saying, "Have mercy on me, Lord, Son of David; my daughter is cruelly demon-possessed."
23. But He did not answer her a word. And His disciples came and implored Him, saying, "Send her away, because she keeps shouting at us." But He answered and said, "I was sent only to the lost sheep of the house of Israel."
25. But she came and began to bow down before Him, saying, "Lord, help me!" And He answered and said, "It is not good to take the children's bread and throw it to the dogs." But she said, "Yes, Lord; but even the dogs feed on the crumbs which fall from their masters' table." Then Jesus said to her, "O woman, your faith is great; it shall be done for you as you wish." And her daughter was healed at once. NASU
From John 5:33-47
33 "You have sent to John, and he has testified to the truth. 34 "But the testimony which I receive is not from man, but I say these things so that you may be saved. 35 "He was the lamp that was burning and was shining and you were willing to rejoice for a while in his light.
36 "But the testimony which I have is greater than the testimony of John ; for the works which the Father has given Me to accomplish -the very works that I do -testify about Me, that the Father has sent Me.
37 "And the Father who sent Me, He has testified of Me. You have neither heard His voice at any time nor seen His form. 38 "You do not have His word abiding in you, for you do not believe Him whom He sent.
39 "You search the Scriptures because you think that in them you have eternal life ; it is these that testify about Me; 40 and you are unwilling to come to Me so that you may have life. 41 "I do not receive glory from men ; 42 but I know you, that you do not have the love of God in yourselves. 43 "I have come in My Father's name, and you do not receive Me; if another comes in his own name, you will receive him. 44 "How can you believe, when you receive glory from one another and you do not seek the glory that is from the one and only God ? 45 "Do not think that I will accuse you before the Father ; the one who accuses you is Moses, in whom you have set your hope. 46 "For if you believed Moses, you would believe Me, for he wrote about Me. 47 "But if you do not believe his writings, how will you believe My words ?"
Yeshua kept the entire Mosaic Law throughout his life, then a very strange thing happened, He was crucified on a cross and Jesus died at the age 33, even though He never sinned, Yeshua did not receive the rewards of keeping all of the Mosaic Laws as applied to Him, instead, Yeshua suffered the punishments required for all Jews whom ever violated the Mosaic Law, and through Yeshua, gentiles whom accept the atonement of Yeshua will also enter the Kingdom That God prepared for His children, and created at the time He created the Cosmos. The Cosmos is a fiery furnace that removes our impurities, and the Kingdom where all His Children 'the sons of Israel will be greatly blessed, and gentiles like me whom accepted 'The Gift' through accepting the blood of the PERFECT LAMB OF GOD, atoning blood of Yeshua.
Upon Yeshua's death, came the fulfilling of the Prophecies of Zechariah Chapter 11:
The burning of Herod's Temple.
11: 1 Open your doors, O Lebanon, That a fire may feed on your cedars. 2 Wail, O cypress, for the cedar has fallen, Because the glorious trees have been destroyed ; Wail, O oaks of Bashan, For the impenetrable forest has come down. 3 There is a sound of the shepherds' wail, For their glory is ruined ; There is a sound of the young lions' roar, For the pride of the Jordan is ruined.
The Diaspora:
4 Thus says the LORD my God, "Pasture the flock doomed to slaughter. 5 "Those who buy them slay them and go unpunished , and each of those who sell them says, 'Blessed be the LORD, for I have become rich !' And their own shepherds have no pity on them. 6 "For I will no longer have pity on the inhabitants of the land," declares the LORD ; "but behold, I will cause the men to fall, each into another's power and into the power of his king ; and they will strike the land, and I will not deliver them from their power."
God cancel's the Mosaic Covenant
7 So I pastured the flock doomed to slaughter, hence the afflicted of the flock. And I took for myself two staffs : the one I called Favor and the other I called Union ; so I pastured the flock. 8 Then I annihilated the three shepherds in one month, for my soul was impatient with them, and their soul also was weary of me. 9 Then I said, "I will not pasture you. What is to die, let it die, and what is to be annihilated, let it be annihilated ; and let those who are left eat one another's flesh."
The Crucifixion of Yeshua
Zecariah 10 I took my staff Favor and cut it in pieces, to break my covenant which I had made with all the peoples. 11 So it was broken on that day, and thus the afflicted of the flock who were watching me realized that it was the word of the LORD. 12 I said to them, "If it is good in your sight, give me my wages ; but if not, never mind !" So they weighed out thirty shekels of silver as my wages. 13 Then the LORD said to me, "Throw it to the potter, that magnificent price at which I was valued by them." So I took the thirty shekels of silver and threw them to the potter in the house of the LORD.
The Diaspora ended in 1947 with God returning the Jews to the promised land, Israel. And God's favor toward His people, The Jews - Israel, has returned. Filling the prophecies of Zecariah Chapters 11-13., defeating the surround nations, repeatedly.
1 The burden of the word of the LORD concerning Israel. Thus declares the LORD who stretches out the heavens, lays the foundation of the earth, and forms the spirit of man within him, 2 "Behold, I am going to make Jerusalem a cup that causes reeling to all the peoples around ; and when the siege is against Jerusalem, it will also be against Judah. 3 "It will come about in that day that I will make Jerusalem a heavy stone for all the peoples ; all who lift it will be severely injured. And all the nations of the earth will be gathered against it. 4 "In that day," declares the LORD, "I will strike every horse with bewilderment and his rider with madness. But I will watch over the house of Judah, while I strike every horse of the peoples with blindness. 5 "Then the clans of Judah will say in their hearts, 'A strong support for us are the inhabitants of Jerusalem through the LORD of hosts, their God.' 6 "In that day I will make the clans of Judah like a firepot among pieces of wood and a flaming torch among sheaves, so they will consume on the right hand and on the left all the surrounding peoples, while the inhabitants of Jerusalem again dwell on their own sites in Jerusalem. 7 "The LORD also will save the tents of Judah first, so that the glory of the house of David and the glory of the inhabitants of Jerusalem will not be magnified above Judah. 8 "In that day the LORD will defend the inhabitants of Jerusalem, and the one who is feeble among them in that day will be like David, and the house of David will be like God, like the angel of the LORD before them. 9 "And in that day I will set about to destroy all the nations that come against Jerusalem.
10 "I will pour out on the house of David and on the inhabitants of Jerusalem, the Spirit of grace and of supplication, so that they will look on Me whom they have pierced ; and they will mourn for Him, as one mourns for an only son, and they will weep bitterly over Him like the bitter weeping over a firstborn.
11 "In that day there will be great mourning in Jerusalem, like the mourning of Hadadrimmon in the plain of Megiddo. 12 "The land will mourn, every family by itself ; the family of the house of David by itself and their wives by themselves ; the family of the house of Nathan by itself and their wives by themselves ; 13 the family of the house of Levi by itself and their wives by themselves ; the family of the Shimeites by itself and their wives by themselves ; 14 all the families that remain, every family by itself and their wives by themselves.
I am a gentile, and I longed that all Israel be saved by YHWH.
Romans Chapter 20 The Apostle Paul we are informs all true Israel will be saved near the end of the age, which is now has has been happening since 1947. For the times for gentiles to enter the Kingdom Is coming to an end, but all Israel, God's first love, will be saved. And I rejoice in this with an open heart.