Goyim is the plural of goy, nation. English usually renders it "gentiles," by way of the Latin gentilis, and over the centuries the word picked up a flavor it does not carry in the Hebrew: outsiders, the unclean, the ones outside the fence. That flavor obscures the point the Tanakh keeps making. The goyim were never the problem to be excluded. They were the purpose to be reached.

Notice first that Israel itself is called a goy. "I will make of you a goy gadol," a great nation, HaShem tells Abraham (Genesis 12:2). The word is not a slur; it simply means a people. But its plural, goyim, came to mean the nations other than Israel, and the covenant's relationship to them was outward from the very first sentence: "in you all the families of the earth will be blessed" (Genesis 12:3). The blessing was always aimed past Israel, through Israel, to everyone else.

The prophets make the aim explicit. "I will make you a light to the goyim, that My salvation may reach to the end of the earth" (Isaiah 49:6). Israel was chosen to carry something outward, not to hoard it inside the fence. So when the most learned Torah scholar of his generation turned to face the Greek-speaking world as a shaliach, a sent one, he was not abandoning his roots. He was the root doing exactly what it was planted to do. The nations are grafted in, not handed a separate tree. The mission to the goyim was Hebraic long before it was ever Greek.