Your idea is exactly what the Calvinists teach, that being "born again" (regeneration) precedes "faith." That is according to the "logical order of salvation" spoken here by Calvinist R.C. Sproul:
"The 'ordo salutis' is the order of salvation. This focuses on the acts of God and the response of the individual in salvation. God calls us, produces regeneration in us, so that we respond with repentance, faith, and obedience. Behind the divine call is God’s electing decree. The 'ordo salutis' is not concerned with a temporal sequence of events, but with a logical order" (The Order of Salvation, Ligonier Ministries, The Teaching Fellowship of R.C. Sproul).
According to this the logical order is first, God calling us. Then secondly, He produces regeneration (born again) in us. Then finally that enables us to respond in faith.
So being "born again" precedes faith in the Calvinist's logical order. But both Peter and James makes it plain that is is faith which results in being "born again":
"Being born again, not of corruptible seed, but of incorruptible, by the word of God...And this is the word which by the gospel is preached unto you" (1 Pet.1:23,25).
"He chose to give us birth through the word of truth, that we might be a kind of firstfruits of all he created" (Jas.1:18).