Paul says that the Gentiles were 1) separated from Christ, 2) alienated (apēllotriōmenoi)) from the commonwealth of Israel, 3) strangers (xenoi) to the covenants, and 4) having no hope and without God (Eph. 2:12). His point, surely, is that all of those have now been overcome. It is fundamental to Paul’s theology that Gentiles have become heirs of the covenant promises by faith (Rom. 4; Gal. 3). He also says that they are no longer strangers (xenoi)—that is, strangers to the covenants of promise—and foreigners (paroikoi) but are “fellow citizens (sympolitai) with the saints and members of the household of God” (Eph. 2:19), all of which is an explicit reversal of the exclusion described in verse 12).
What has changed is that Jesus’ death has removed the dividing partition of the Law which required male Gentiles to become circumcised Jews if they wished to be part of the commonwealth of Israel. If the point wasn’t inclusion in a renewed Israel, the abolition of the Law would be irrelevant. The argument elsewhere with Judaizers was that Gentiles were already members of Israel-undergoing-renewal by virtue of the Spirit, therefore Law-observance was unnecessary.
This is what I mean when I say that the underlying assumption in the New Testament is that Gentiles were becoming part of a restored Israel—under a crucified king and in the power of the Spirit of the new covenant that was promised to Israel by the prophets (cf. Eph. 2:20-22). Much of the time it simply goes unstated because it was the obvious premise: the Jewish apostles were proclaiming a Jewish Lord and Messiah, a Jewish king to rival Caesar.
As Paul says in Romans 15:8-12, Jesus became a servant to the circumcised in order to confirm the promises to the patriarchs so that in the long run the “root of Jesse” (ie., Israel’s king) would rule over the nations. The restoration of Israel leads to a new Jewish empire.