While the angel could have given Cornelius the Gospel directly, he did not have the mandate from the Lord to do that. His role was to do what the Lord instructed. The mandate was for the human disciples of Christ to preach the Gospel, and that Peter, being the most available person near to where Cornelius lived, he would be the most appropriate. As it was said in this thread already, it was just as much for Peter and the other Apostles that it happened this way. If the angel had given the Gospel directly, there would be no outside human witnesses that the Gospel had come to the Gentiles as well as the Jews, because up until that time, the Gospel was preached to the Jews only. Therefore, the intention was that Peter goes and preaches the Gospel to the Gentile household, witnesses the Holy Spirit falling on them in the same way that He fell on the disciples at Pentecost, and then for Peter and his companions to go to Jerusalem to testify to the other Apostles what happened.
The rule for us is that we are given the commission to preach the Gospel to every person and this is why our testimony of what Christ as done for us is vitally important, and that because people read us before they read the Bible, holiness of life and genuineness of our faith is essential to show that being a Christian believer is something very significant in our lives.
In the parable of the sower, Jesus described preaching the gospel as broadcasting seed, rather than planting one seed at a time. As a result of broadcasting the seed, some would fall on hard ground, some in rocky soil, some among weeds and some in good soil.
By contrast, those who live in apartments or small houses, plant one seed at a time in good soil, hoping to have tomato plants, flowers, or what-have-you. Unlike broadcasting the seed, which falls on various kinds of soils, the apartment dweller already knows in advance that the soil is good and that the single seed will take root and grow to maturity.
Isn't it interesting that, in this case, Cornelius was instructed to send for Peter
and Peter was instructed to visit Cornelius? And isn't Peter's visit to a small household more like the apartment dweller planting seed in good soil, than the farmer broadcasting seed? And just as the apartment dweller knows the soil is good and productive in advance of planting the seed, what do you think God knew about the heart of Cornelius, which had to be true before he sent Peter to preach the gospel to this man and his family personally?
According to the parable of the sower, there is no guarantee that the seed will take root or produce a crop. In other words, the preacher has no guarantee that everyone hearing the gospel will believe it or persevere. But, in the case of Peter and Cornelius? Isn't it logical to conclude that God had already prepared the heart of Cornelius to hear and accept the gospel, and that upon hearing it would believe it unto salvation?
Just thinking out loud here.
@Tong2020