Why did God save the house of Cornelius according to Les?

God did want the house of Cornelius saved, but the larger purpose was to defend Paul at the Jerusalem council. Peter's memory of Cornelius allowed Paul to return to Antioch with freedom to preach his Gospel, preventing the Jews from hindering him every step.

Les explains this as the sovereign working of a sovereign God. Yes, God wanted the house of Cornelius absolutely, but the bigger purpose was to come to Paul's defense in the council at Jerusalem. If Peter had not remembered what happened at the house of Cornelius, they never would have let Paul go back to Antioch with freedom to preach his Gospel. They would have plagued him every inch of the way. Peter realized that twelve years earlier God had shown him He could save Gentiles without going through all that they demanded, and that testimony protected Paul's freedom to preach.

Acts 15:7

And when there had been much disputing, Peter rose up, and said unto them, Men and brethren, ye know how that a good while ago God made choice among us, that the Gentiles by my mouth should hear the word of the gospel, and believe.

What would have happened to Paul if Peter hadn't remembered the house of Cornelius?

Had Peter not recalled Cornelius’s household, the Jerusalem council would not have permitted Paul to return to Antioch with freedom to preach his Gospel. Instead, they would have pursued and plagued him at every step, opposing his ministry among the Gentiles.