Depravity in my view does not teach helplessly depraved the way I take you to mean it.
Can you show me using Scripture that Cornelius was not already saved? Was the pouring out
of the Holy Spirit not a demonstration to show Peter that God had opened the way to Gentiles?
Does that necessarily mean he did not already have the Holy Spirit? Or that his heart was not
circumcised? Do you believe Cornelius was a still a natural man in all aspects?
I can show he wasn't already saved yes.
Peter tells the story of what happened in Acts 11:13-14:
He told us how he had seen an angel appear in his house and say, ‘Send to Joppa for Simon who is called Peter.
He will bring you a message through which you and all your household will be saved.’
They will be saved thru the message Peter brought, not prior to it!
The reason why Peter was sent to Cornelius was because he was a just man and God had heard his prayers and his alms have come to God's knowledge, he was a God-fearing gentile, respected among the jews as well I would presume.
God wanted to get this man saved, and He did, He sent Peter. This also served as a lesson to Peter that God is in the business of saving Gentiles, it also served as a lesson for everyone that God doesn't show favoritism but accepts from every nation the one who fears him and does what is right.
Some of the gentiles in the churches that Paul is writing to were these "God-fearing gentiles" that were hanging around the jews and learning alot from them, but just didn't officially convert to judaism. They were the first ones to convert, why? Because they were already "taught by the Father" in that sense, so to them, accepting Christ was just something that naturally followed. Same for the apostles and all the jews who were already part of the remnant of Israel, its near guarantee they will believe in Jesus, the scribes and pharisees? Not as likely.