I don't think anyone is saying they don't have ability to choose.
Regarding salvation, Calvinists are saying exactly that.
Choosing though does not clinch salvation
Faith does.
Once a person has heard the gospel, that person has the free will ability to choose to believe it, and once he does, he becomes saved.
12) But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:
13) Which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God.
"which were" in v13 should be "who was" and it refers back to "his name" in v12. Jesus Christ was born, not of bloods (it's plural in the Greek), nor of the will of flesh or man, but of God. IOW God sent Jesus Christ into the world by HIS will and not man's will.
James 1:
18) Of his own will begat he us with the word of truth, that we should be a kind of firstfruits of his creatures.
The verse is not stating that God decided which individual people He would beget, but it is speaking collectively: God decided to beget Christians. That happens when people hear the gospel and decide to believe it, something anyone can do.
1 Pet 1:
3) Blessed be the God and Father of our Lord Jesus Christ, which according to his abundant mercy hath begotten us again unto a lively hope by the resurrection of Jesus Christ from the dead,
This should be understood the same as James 1:18. The "us" is collective, not individual.
Salvation isn't a choice nor is it caused by a choice.
Salvation is a choice and it is initiated by choice.
God wants all men to be saved and come to the knowledge of the truth (1 Tim 2:4). He is not willing that any should perish, but that all should come to repentance (2 Pet 3:9).