My point of view is clear in the texts you have already cited.
I wasn't born into the new covenant, I was adopted into it as an adopted child of God through salvation and faith in Christ. This is basic Christianity.
Here are some more verses to add to the ones you have already cited:
John 1
9 The true light that gives light to everyone was coming into the world. 10 He was in the world, and though the world was made through him, the world did not recognize him. 11 He came to that which was his own, but his own did not receive him. 12 Yet to all who did receive him, to those who believed in his name, he gave the right to become children of God— 13 children born not of natural descent, nor of human decision or a husband’s will, but born of God.
And one more:
1 John 1
8 If we claim to be without sin, we deceive ourselves and the truth is not in us. 9 If we confess our sins, he is faithful and just and will forgive us our sins and purify us from all unrighteousness. 10 If we claim we have not sinned, we make him out to be a liar and his word is not in us.