Paul talks about how it's the Christians who are the truly circumcised people because they have made themselves right with God. So, in the context of the new covenant, are the Christians now God's chosen people?
Let’s assume that “Christian” refers only to those who are truly saved.
First, Christians have not made themselves right with God. Jesus Christ has made them right with God by his life, death, and resurrection. Circumcision of the heart is by the Spirit (Rom 2:29; Deut 30:6), and righteousness is received, not self-produced (Php 3:9; Isa 64:6). And yes, he did this because they are God’s chosen people (elect).
Second, the Israel of God (Gal 6:16) has never been defined by ethnicity; it has always consisted of those inside and outside ethnic Israel, so believing Gentiles do not displace his people (cf. Isa 56:6-7). Scripture distinguishes between ethnic Israel and the elect within Israel (Rom 9:6-8). The Abrahamic promise—indeed, even the Edenic promise (Gen 3:15)—was always teleological, the focus being the Christ and all those chosen in him (Eph 1:4; Gal 3:16, 26-29). The promised seed is and has always been Christ and those he represents. This is covenantal fulfillment, not ethnic displacement (Acts 15:14-18; cf. Rom 11:25). The church is the expansion and fulfillment of Israel in Christ; the olive tree remains one (Rom 11), as it always has: Unbelieving Jews are broken off, believing Jews remain, and believing Gentiles are grafted in. One people of God, one root, one covenantal structure, one faith, a cruciform promise fulfilled in Christ.
Third, one should keep in mind that this is how God has always worked. This pattern of judgment, exile, and restoration is repeated throughout the Old Testament. Israel is unfaithful (unbelief), they are judged and exiled (broken off), a faithful remnant is spared (remain in the olive tree), and there is a promise of restoration (grafted back in).
From the Old Testament to the New, there has always been a visible covenant people within which are God’s chosen people, the elect united to Christ by the Spirit in due time.