New Testament salvation implies remission of sins.
When you think of New Testament salvation, you think of complete and thorough forgiveness/remission of sins. That theme was first announced in prophecy (e.g., Jeremiah 31:31-34; Micah 7:19; Ezekiel 36:25-29). It was proclaimed by John the Baptist who introduced Christ and His mission (John 1:29). It’s implied in Jesus going to the cross to shed His blood for our sins. He came as the world’s Savior, which implies salvation from sin. When He walked on this earth, He ministered, saying, “Thy sins be forgiven thee” (Mark 2:5). In fact, the very name of Jesus is associated with remission of sins. He was so named by the angel because He would save His people from their sins (Matthew 1:21). Peter declared to Cornelius that all the prophets witnessed that remission of sins would come through His name (Acts 10:43). And then Peter baptized Cornelius and his household in the name of Jesus (v.48). Scripture is perfectly clear that baptism in Jesus’ name is distinctly for the remission of sins (Acts 2:38; 22:16).
Jesus’ name and baptism are both associated with remission of sins. Where does Matthew 28:19 come in?
Consider Matthew’s purpose for writing his Gospel account. It was all about Jesus. He made that clear from the very first verse (Matthew 1:1). The book is about the coming forth of Jesus Christ. And, immediately, Matthew presented Jesus as the Jewish God. After showing how the angel associated the name of Jesus with remission of sins (Matthew 1:21), he then used the well-known quotation of Isaiah 7:14 to point out that Jesus is, without qualification, the incarnation of God. Matthew was not going to change his stated theme at the end of his book. He didn’t conclude his Gospel account by presenting the titles of Father, Son, and Holy Ghost as the formula for remission of sins. His closing with Matthew 28:19 rounded out his declared theme and showed remission of sins to be associated with the name of Jesus in a profound way. The words of Christ in Matthew 28:19 were Jesus’ assertion of His full and absolute deity— that He is the incarnation of the Father. Among the Gospel writers, Matthew is especially noted for pointing out the fulfillment of Messianic prophecies because he had Jewish readers in mind. As mentioned, he began his Gospel by doing so with the prophecy of Isaiah 7:14. Many others followed. When he quoted Jesus in Matthew 28:19, he effectively showed the fulfillment of the twin Isaiahan prophecy revealing the identity of Christ, namely, Isaiah 9:6. In his Gospel, Matthew was inspired to present Jesus as the fulness of the Jewish God and that remission of sins comes through His name, which is exactly how Matthew 28:19 was fulfilled on the Day of Pentecost (Acts 2:38-39).