He actually did. The baptism in Jesus’ name isn’t a formula, but a shorthand way to distinguish the baptism Jesus gave (in the name of the Father and the Son and the Holy Spirit) with the baptism of John.
One way we can know that this baptism was actually the Trinitarian form is looking at Acts 19. There St. Paul finds some believers in Ephesus, and he asks them if they have received the Holy Spirit yet. They tell him “We have never even heard of the Holy Spirit.” Paul then asks, “In what, then were you baptized?” (Acts 19:3) They tell him they were baptized with John’s baptism, and then Acts says Paul baptized them all in Jesus’ name.
Notice though, when they said they never heard of the Holy Spirit, Paul immediately asks them what baptism they received, and then he baptizes them in the name of Jesus. If the name of Jesus was the actual formula used, why would Paul seem confused about which baptism the Ephesians had received? Why would that make a difference?
Because, if they had been baptized with Jesus’ baptism, they would have heard of the Holy Spirit, because the Holy Spirit is invoked in that baptism. That’s the formula Jesus himself gave us and that’s the formula the Church has used from the beginning.