The believing in Christ here is the result of the Fathers Drawing and teaching. They could not believe without being taught of Christ.
It is the result of believing God's word.
John 6:45 (NASB 95) — 45 It is written in the prophets, ‘AND THEY SHALL ALL BE TAUGHT OF GOD.’ Everyone who has heard and learned from the Father, comes to Me.
Christ first has to spiritually revealed in us and to us before we believe in Him. Like He was revealed in Paul Gal 1:16
To reveal his Son in me, that I might preach him among the heathen; immediately I conferred not with flesh and blood:
Peter by nature did not know who Christ was, it was supernaturally revealed to him Matt 16:16 And Simon Peter answered and said, Thou art the Christ, the Son of the living God.
17 And Jesus answered and said unto him, Blessed art thou, Simon Barjona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven.
Again Matt 11:27
27 All things are delivered unto me of my Father: and no man knoweth the Son, but the Father; neither knoweth any man the Father, save the Son, and he to whomsoever the Son will reveal him.
So by nature we can not believe in Christ. We can not know Him by nature, no man does.
The real issue is can man respond to information about Christ and believe
In your quotation
Matthew 16:16–17 (NASB 95) — 16 Simon Peter answered, “You are the Christ, the Son of the living God.” 17 And Jesus said to him, “Blessed are you, Simon Barjona, because flesh and blood did not reveal this to you, but My Father who is in heaven.
Peter believed the revelation concerning Christ
So despite your previous claim that one is first regenerated and then believes we have no such claim
Those who do not believe will not see life
John 3:36 He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him.
John 3:18 The one who believes in him is not judged, but the one who does not believe has already been judged, because he has not believed in the name of the one and only Son of God.
Mark 16:16 The one who believes and is baptized will be saved, but the one who refuses to believe will be condemned.
John 8:24 Thus I said to you that you will die in your sins. For if you do not believe that I am he, you will die in your sins.”
1 John 5:12 The one who has the Son has the life; the one who does not have the Son of God does not have the life.
1 John 5:10–13 (The one who believes in the Son of God has the testimony in himself. The one who does not believe God has made him a liar, because he has not believed in the testimony that God has testified concerning his Son.) And this is the testimony: that God has given us eternal life, and this life is in his Son. The one who has the Son has the life; the one who does not have the Son of God does not have the life. These things I have written to you who believe in the name of the Son of God, in order that you may know that you have eternal life.
So scripture refutes Calvinism’s belief that unbelievers may be given life so that they may believe
That those who believe will attain eternal life is seen below
John 1:12–13 (NASB 95) — 12 But as many as received Him, to them He gave the right to become children of God, even to those who believe in His name, 13 who were born, not of blood nor of the will of the flesh nor of the will of man, but of God.
John 20:30–31 (NASB 95) — 30 Therefore many other signs Jesus also performed in the presence of the disciples, which are not written in this book; 31 but these have been written so that you may believe that Jesus is the Christ, the Son of God; and that believing you may have life in His name.
John 3:15–16 (NASB 95) — 15 so that whoever believes will in Him have eternal life. 16 “For God so loved the world, that He gave His only begotten Son, that whoever believes in Him shall not perish, but have eternal life.
John 6:40 (NASB 95) — 40 For this is the will of My Father, that everyone who beholds the Son and believes in Him will have eternal life, and I Myself will raise him up on the last day.”
Proven here also
John 5:40 (ESV) — 40 yet you refuse to come to me that you may have life.