I like what William Kelly says here, in his Commentary on Matthew 10 :
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"Now chapter 10 shows that Jesus, as Lord of the harvest, sends forth labourers and this too with full authority and power given to them. But observe, it is still in special connection with Israel; and the Lord is conscious from the beginning of rejection by Israel. Meanwhile it is a Jewish mission of the twelve Jewish apostles to the lost sheep of the house of Israel. I take this quite literally, and not as if it were said of the Church, which is never spoken of as lost sheep; but the sheep of Israel in their desolate condition are most aptly so described. Before the Church is gathered, what we want is a Saviour. We Gentiles were not sheep at all, but dogs, in our Evangelist's point of view. (See Matt. 15) And after we have been brought into the Church, we are not, and cannot be, lost sheep. Whereas these poor of the flock are spoken of as lost sheep of the house of Israel. For up to this time the work was not done by which they could be put in the known position of salvation.
"Again, when our Lord is sending them forth, it is said, "He called unto Him His twelve disciples, and gave them power against unclean spirits, to cast them out, and to heal all manner of sickness, and all manner of disease" (ver. 1). This was peculiarly their mission. Not a word is said about preaching what we call the gospel, or teaching the whole counsel of God; but they were to go with messianic power against Satan and bodily diseases, as a testimony to Israel. They were to declare the kingdom of heaven. "As ye go," said our Lord, "preach, saying, The kingdom of heaven is at hand" (ver. 7). But the great characteristic feature of the mission was the conferring upon them power against demons and diseases. The appropriateness of this, in connection with Israel, is manifest. It was a bright evidence that the true King, Jehovah, was there, who was able Himself not only to cast out devils, but to confer that power upon His servants. Who but the King, the Lord of hosts, could do this? It was a testimony much greater than if the power had been confined to His own person. The ability to impart power to others (which was what Simon Magus, hoping to profit by it, so earnestly coveted) God here shows to be in His own Son. Now the servants were to be sent out, and in due order - twelve of them, in relation to the twelve tribes of the house of Israel. We find afterward the promise that they should "sit upon twelve thrones, judging the twelve tribes of Israel." There need be no question, therefore, that this was a Jewish mission. When the Church was called, God broke in upon the mere Jewish order by calling an extraordinary apostle, with a special view to the Gentiles - one who was called after Christ had died, and risen, and had taken His place at the right hand of God. Then came in this new work in the calling of the Church, and the apostle Paul became the characteristic minister of the Church, though the twelve had their place too. But at this time the twelve apostles were to be (what Paul was not) the ministers to Israel in testimony of the kingdom of heaven. For, observe, the strictest injunction was given them that they were not to go outside the limits of Israel; not even to visit the Samaritans, nor to enter into the cities of the Gentiles. Their business was solely with the lost sheep of the house of Israel: a positive proof that it means those of the Jews who had a sense of sin, and who were willing to receive the testimony of the true Messiah. With them, their business was exclusively. It is the more remarkable, because in this Gospel we are told that after He had died and was risen, the Lord sent them out to the Gentiles; but then it was on the evident ground that His death had come in. "I, if I be lifted up, will draw all men unto Me." Christ upon the cross becomes the attractive centre for man, as well as the foundation of all the counsels of God. Now in this case we have nothing of the sort. The Lord's death is not even referred to. His rejection is brought out, but nothing is said as to the building of a new structure - the Church. There was the waiting for still further rejection before this could be disclosed, as in Matthew 16.
"But here the Lord Jesus sends forth the twelve, and commands them, saying, "Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not: but go rather to the lost sheep of the house of Israel. And as ye go, preach, saying, The kingdom of heaven is at hand. Heal the sick, cleanse the lepers, raise the dead, cast out devils: freely ye have received, freely give. Provide neither gold, nor silver, nor brass, in your purses; nor scrip for your journey, neither two coats, neither shoes, nor yet staves: for the workman is worthy of his meat" (vers. 5-10). That is, they were to go just as they were, with the coat they had upon them, with the shoes they had then on their feet. They were not to provide anything, or to lay up any store as a means of support during their mission. This is not a universal rule for the servants of God at all times. It was a peculiar mission, for a special time, and with reference to Israel only. It was not the gospel of God's grace, but of the kingdom. The two go together now; but then it was not so. Israel did not receive the testimony of the kingdom; an entire change comes in, and the kingdom of heaven, in outward establishment, remains in suspension. The calling of God now to the Gentiles comes in as a vast parenthesis between this message to the lost sheep in Israel and its full accomplishment in the last days. Whatever the Lord commands must be accomplished, but nothing is perfectly fulfilled till the Lord takes all in hand Himself."
--William Kelly, Commentary on Matthew 10 [source: BibleHub]
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