One of the major components of OT religion is the Temple. It is an essential element of Jewishness. It factors also into the millennial schemes of some. While Dispensationalists would reject the notion of future sacrifices, a Restored Temple is a future expectation for them.
In contrast, the biblical writers viewed that there was an essential transferal of OT theology from national Israel to the Church. This was not so much a matter of making the Church equivalent to Israel, but rather that the Church was included into Israel by virtue of Christ's position as heir of the OT promises. As such, the Church is the true remnant of Israel, being akin to the stones being raised up as children for Abraham on the basis that they, like Abraham, believed in God.
One example of the transferal of OT theology to the church is Temple theology. The essential element of Temple theology is that the Temple was the locus of God's presence. Without this element, Temple theology would be nil. The motif is so strong and obvious, I won't detail it in this overview, although I wrote about 150 pages on it in my Master's thesis.
Matthew takes this motif and transfers it to the church. In his view, the Temple was no longer the locus of God's presence. Like Ezek 8-11, Matthew represents Jesus Immanuel as abandoning the Temple at the conclusion of Matt 23 and Matt 24:1. Rather than God's presence residing at the Temple, it has moved to reside in God's people, Jesus' disciples.
This portion of Matthean theology is perceptible in his Immanuel inclusio. Jesus himself is Immanuel, God-with-us (Matt 1:21). And he is with us always, even to the consummation of the age (Matt 28:20. Moreover, the theme is reinforced at significant points in Matthew, for where two or three are gathered in Jesus' name, Immanuel is present with them.
The fact that God's presence is removed from the Temple, makes the Temple a haunt for every kind of foul spirit and abomination; the swept house is made seven times worse. It is left to them desolate. It deserves to be torn down, without a stone being left atop the other. No longer is it a house of prayer, as the house of prayer now belongs to the disciples of Jesus who gather in his name, and among whom Immanuel is present as they ask anything in his name.
In every conceivable way, Matthew takes every aspect of Temple theology and transfers it to the church. Much more could be said.
But the one point I'm arguing is that the Church receives all the OT promises. Temple theology is but one example of an OT theme or promise being transferred to the Church. This was an interpretive matrix of the earliest Christians. By virtue of their faith in Jesus—the Supreme Jew—they were all made joint-heirs with Jesus, regardless of race.