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The question may be answered from a general Christian perspective since Catholics and Protestants are not divided on the "catholicity" issue as pertains to the early church. The church is for everyone, Jews and Gentiles, slaves and freemen, saints and sinners.
Prior to the episode mentioned in the OP, in Chapter 10:5-6 of Matthew's gospel we find the following:
‘Go nowhere among the Gentiles, and enter no town of the Samaritans, but go rather to the lost sheep of the house of Israel.'
Don't 'go' to Gentiles, but do minister when asked
However, in chapter 14, quoted in the OP, even though he has not been "sent" to Gentiles, Jesus is convinced to heal a Gentile woman.
Just then a Canaanite woman from that region came out and started shouting, ‘Have mercy on me, Lord, Son of David; my daughter is tormented by a demon.’ But he did not answer her at all. And his disciples came and urged him, saying, ‘Send her away, for she keeps shouting after us.’ He answered, ‘I was sent only to the lost sheep of the house of Israel.’ But she came and knelt before him, saying, ‘Lord, help me.’ He answered, ‘It is not fair to take the children’s food and throw it to the dogs.’ She said, ‘Yes, Lord, yet even the dogs eat the crumbs that fall from their masters’ table.’ Then Jesus answered her, ‘Woman, great is your faith! Let it be done for you as you wish.’ And her daughter was healed instantly.
Here, Jesus equates Gentiles with dogs and at first declines to help the woman. He is moved by her response and cures her daughter accordingly.
All three synoptic gospels also include the story of the centurion's slave, whom Jesus heals after the centurion's humble expression of faith. Jesus marvels that such faith was not found in Israel, and he heals the slave without even seeing him.
‘Lord, my servant is lying at home paralyzed, in terrible distress.’ And he said to him, ‘I will come and cure him.’ The centurion answered, ‘Lord, I am not worthy to have you come under my roof; but only speak the word, and my servant will be healed. .." When Jesus heard him, he was amazed and said to those who followed him, ‘Truly I tell you, in no one in Israel have I found such faith. I tell you, many will come from east and west and will eat with Abraham and Isaac and Jacob in the kingdom of heaven. (Mt. 8)
Here the exclusivity of Jesus sayings in other passages is universalized. Not only does he heal the slave when asked to by his master, he teaches that people from all over the world will sup with with the patriarchs. This set the stage for what would happen later as the church began to actively evangelize among non-Jews.
Moreover, in the Gospel of John (ch. 4), Jesus even leads his disciples personally into a Samaritan town and witnesses to a Samaritan woman. In Luke 8, he tells the story of the good Samaritan, showing that even this semi-Gentile group can be better "neighbors" than priests and Pharisees if they show compassion to others.
Thus there was already ample reason for the Church to minister to non-Jews, at least when invited to do so.
The Great Commission
However, the most compelling justification for "going" to non-Jews with the good news of the Gospel is found in the Great Commission:
‘All authority in heaven and on earth has been given to me. Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 and teaching them to obey everything that I have commanded you.' (Mt. 28:18-20)
Once the Great Commission was issued, the Gospel could be preached universally. In evangelizing, there was no longer any need to distinguish between Jews and non-Jews or to make distinctions between Hebrews, Samaritans, Israelites and even Gentiles. However, problems would arise as to whether Gentile believers needed to adopt Jewish traditions such as circumcision and keeping kosher. These would be resolved through the eventual acceptance of the teachings of Paul (see below).
The Early Church
The first missions to non-Jews are already mentioned in Acts 8:1. They were conducted at least partly out of necessity.
A severe persecution began against the church in Jerusalem and all except the apostles were scattered throughout the countryside of Judea and Samaria. [Samaria was the land of the Samaritans, between Judea and the Galilee.]
This took place when the future apostle Paul was still persecuting the church. After his conversion, Paul was eventually designated "apostle to the Gentiles." Samaritans and Gentiles had already become believers, but Paul preached forcefully that there should be not distinctions based on nationality; and in his teaching they did not need to adopt Jewish customs.
In Christ Jesus you are all children of God through faith. As many of you as were baptized into Christ have clothed yourselves with Christ. There is no longer Jew or Greek, there is no longer slave or free, there is no longer male and female; for all of you are one in Christ Jesus. And if you belong to Christ, then you are Abraham’s offspring, heirs according to the promise. (Gal. 3:26-29)
Eventually the Pauline doctrine prevailed. Vestiges of specifically exclusively Jewish Christianity would remain, but the command to "preach the Gospel to all nations" would take precedence over any appeal to tribalism.
Note: something to add to the second paragraph of the OP -- In the Old Testament scriptures, "Israel" often describes the northern tribes and the Kingdom of Israel. "Judah" describes the southern tribes and the Kingdom of Judah. "Ephraim" sometimes refers to the northern kingdom as well, since it was the dominant northern tribe. The term "Jews" applied specifically to those Judahites who returned from exile to Judea. Eventually the word would encompass Israelites of every tribe.
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You have to go back to the Book of Genesis.
In chapter 12 God says: 3 And I will bless them that bless thee, and curse him that curseth thee: and in thee shall all families of the earth be blessed.
"Niberechu" (נִבְרְכ֣וּ) translated as blessed can also be legitimately translated as grafted. "In your seed [Jesus (Galatian 3:16)] shall all families of the earth be grafted."
ונברכו בך כל משפחות האדמה, “and all the families of the earth shall count themselves blessed if they can establish family ties with you.” The root of the word ונברכו is the same as that of מבריך, “to graft.” (Daat Zkenim) https://www.sefaria.org/Genesis.12.3?lang=bi&with=Daat%20Zkenim&lang2=en
In Genesis 17 God added: 4 As for me, behold, my covenant is with thee, and thou shalt be a father of many nations. 5 Neither shall thy name any more be called Abram, but thy name shall be Abraham; for a father of many nations have I made thee. 6 And I will make thee exceeding fruitful, and I will make nations of thee, and kings shall come out of thee.
In Genesis 48 Jacob adopts two sons of Joseph as his own:
13 And Joseph took them both, Ephraim in his right hand toward Israel's left hand, and Manasseh in his left hand toward Israel's right hand, and brought them near unto him. 14 And Israel stretched out his right hand, and laid it upon Ephraim's head, who was the younger, and his left hand upon Manasseh's head, guiding his hands wittingly; for Manasseh was the firstborn. 15 And he blessed Joseph, and said, God, before whom my fathers Abraham and Isaac did walk, the God which fed me all my life long unto this day, 16 The Angel which redeemed me from all evil, bless the lads; and let my name be named on them, and the name of my fathers Abraham and Isaac; and let them grow into a multitude in the midst of the earth. 17 And when Joseph saw that his father laid his right hand upon the head of Ephraim, it displeased him: and he held up his father's hand, to remove it from Ephraim's head unto Manasseh's head. 18 And Joseph said unto his father, Not so, my father: for this is the firstborn; put thy right hand upon his head. 19 And his father refused, and said, I know it, my son, I know it: he also shall become a people, and he also shall be great: but truly his younger brother shall be greater than he, and his seed shall become a multitude of nations.
In verse 19 Jacob says that Ephraim shall become "melo hagoim" (מְלֹא-הַגּוֹיִם) which should be better translated as "fulness of the gentiles".
The crossed hands of Jacob prophetically tell that the gentiles would be grafted to the family of God through the cross.
Paul in the Letter to Romans talks about both, grafting and the fullness of the gentiles in chapter 11. He understood.
After Salomon's death, the kingdom split in two. The ten northern tribes called Israel, Joseph or often Ephraim were eventually exiled by the Assyrians and dispersed. There they mingled with other tribes and nations and lost their identity. They couldn't be physically distinguished as descendants of Abraham. But because of the dispersion, every gentile believer in Jesus could be considered a spiritual descendant of Abraham through Ephraim. These are the lost sheep of Israel. John hints this in his gospel:
54 Jesus therefore walked no more openly among the Jews; but went thence unto a country near to the wilderness, into a city called Ephraim, and there continued with his disciples.
All the prophecies of the OT considering unification of Judah and Ephraim, like Ezekiel 37, pictured reunion of the Jews and the gentiles in Christ.
Another important clue from the OT is Isaiah 56: 6 Also the sons of the stranger, that join (הַנִּלְוִים – from the same root name Levi is derived; Isaiah prophecied that gentiles would be God's priests) themselves to the LORD, to serve him, and to love the name of the LORD, to be his servants, every one that keepeth the sabbath from polluting it, and taketh hold of my covenant; 7 Even them will I bring to my holy mountain, and make them joyful in my house of prayer: their burnt offerings and their sacrifices shall be accepted upon mine altar; for mine house shall be called an house of prayer for all people.
The only time when Jesus was angry happened in the temple. The court of the gentiles was transformed into a market, which made worshipping God by believing foreigners impossible. In Mark 11:17 Jesus explains Himself quoting Isaiah 56.
Also Isaiah 66: 18 For I know their works and their thoughts: it shall come, that I will gather all nations and tongues; and they shall come, and see my glory. 19 And I will set a sign among them, and I will send those that escape of them unto the nations, to Tarshish, Pul, and Lud, that draw the bow, to Tubal, and Javan, to the isles afar off, that have not heard my fame, neither have seen my glory; and they shall declare my glory among the Gentiles. 20 And they shall bring all your brethren for an offering unto the LORD out of all nations upon horses, and in chariots, and in litters, and upon mules, and upon swift beasts, to my holy mountain Jerusalem, saith the LORD, as the children of Israel bring an offering in a clean vessel into the house of the LORD. 21 And I will also take of them for priests and for Levites, saith the LORD. 22 For as the new heavens and the new earth, which I will make, shall remain before me, saith the LORD, so shall your seed and your name remain. 23 And it shall come to pass, that from one new moon to another, and from one sabbath to another, shall all flesh come to worship before me, saith the LORD. 24 And they shall go forth, and look upon the carcases of the men that have transgressed against me: for their worm shall not die, neither shall their fire be quenched; and they shall be an abhorring unto all flesh.
Also Jeremiah 12: 14 Thus saith the LORD against all mine evil neighbours, that touch the inheritance which I have caused my people Israel to inherit; Behold, I will pluck them out of their land, and pluck out the house of Judah from among them. 15 And it shall come to pass, after that I have plucked them out I will return, and have compassion on them, and will bring them again, every man to his heritage, and every man to his land. 16 And it shall come to pass, if they will diligently learn the ways of my people, to swear by my name, The LORD liveth; as they taught my people to swear by Baal; then shall they be built in the midst of my people. 17 But if they will not obey, I will utterly pluck up and destroy that nation, saith the LORD.
Also Amos 9: 11 In that day will I raise up the tabernacle of David that is fallen, and close up the breaches thereof; and I will raise up his ruins, and I will build it as in the days of old: 12 That they may possess the remnant of Edom, and of all the heathen, which are called by my name, saith the LORD that doeth this.
And many more.