Looking at Commentaries can help. "Pulpit Commentary" is a favorite of mine.
Matthew 10:5 biblehub
Matthew 10:5 Commentaries: biblehub
Pulpit Commentary
*snip*
(1) That it was only right that the proclamation of the coming of Messiah should be thoroughly made to the Jews first. Had they accepted it, they would have become the great factors in the evangelization of the Gentiles (cf.
Romans 11:12, 15); as they rejected it, it was necessary that the offer should, apart from them, be made to others (
Acts 28:28). ...........................................
(2) The apostles were as yet in no fit state spiritually to carry the message beyond their own nation, and the facts which they were in a position to proclaim might, when proclaimed alone, have proved a stumbling-block to the after-acceptance by Gentiles and Samaritaus of a fuller and therefore truer message (cf.
Matthew 28:18, sqq.;
Acts 1:8). Therefore they are now bid perform their present duty without turning away from it, and, as we may add, will-out anticipating their entrance upon a wider sphere. Saying, Go not. This would be outside your course (ἀπέλθητε). In the Greek, however, the following words receive the emphasis. Into the (any, Revised Version) way of the Gentiles (εἰς ὁδὸν ἐθνῶν).
Footnotes
(1) These words are generally understood to mean "into any road that would lead to Gentile lands or districts." So Tyndale, "Go not into the wayes that leade to the gentyls." (For this genitive of direction, cf.
Matthew 4:15;
Jeremiah 2:18, and perhaps, Judith 5:14.)
(2) Weiss, 'Matthaus-ev.,' takes them as equivalent to "into any street in a heathen land," making the genitives, ἐθνῶν and Σαμαρειτῶν, both possessive. There are serious objections to these two interpretations; to the first, that the genitives are then used in different senses; to the second, that it suggests something altogether outside the Israelitish border.
(3) Is not a third interpretation possible - to consider flint our Lord had in his mind the parts of towns, otherwise Jewish, which were inhabited by heathen, just as, in the days of Omri and Ahab, such parts were assigned to Syrians in Samaria, and to Israelites in Damascus, or in modern times to Jews in Christian towns? We have not, indeed, direct evidence of Gentiles, during the time of our Lord, thus living in separate streets, but with the Jewish aversion to even letting them houses and to having more to do with them than possible (cf. Schurer, II. 1:51-56), it would seem probable that, without any formal arrangement being made, the result would be separation of this kind. It is true that ὁδός is not used elsewhere in this sense in the New Testament, but a comparison of passages in the LXX. seems to justify our so interpreting it........................
===============
Barnes' Notes on the Bible
Into the way of the Gentiles - That is, among the Gentiles, or nowhere but among the Jews. The full time for preaching the gospel to the Gentiles was not come. It was proper that it should be first preached to the Jews, the ancient covenant people of God, and the people among whom the Messiah was born. Afterward he gave them a charge to go into all the world,
Matthew 28:19.
===================
Gill's Exposition of the Entire Bible
These twelve Jesus sent forth,...
saying, go not into the way of the Gentiles; meaning, not the customs' and manners of the Heathens, they were to avoid; but that they were not to steer their course, or take their journey towards them: they were not, as yet, to go among them, and preach the Gospel to them; the calling of the Gentiles was not a matter, as yet, so clearly revealed and known, nor was the time of their calling come: besides it was the will of God, that the Gospel should be first preached to the Jews, to take off all excuse from them, and that their obstinacy and perverseness in rejecting Jesus as the Messiah, might manifestly appear; and since Christ himself was the minister of the circumcision, he would have his apostles, for the present, whilst he was on earth, act agreeably to the character he bore, that there might be an entire harmony in their conduct.