Did you not read the last paragraph, The Galatians verses were used in order to demonstrate Adoption.
"The Adoption is only able to take place during the Believers Baptism, which is the time of Re-Generation, the becoming New of all things pertaining to a life in the Spirit."
Adoption does not take place when one believes, but only through the Physical Process of Baptism, where the Believer Physically identifies with Christ's Crucifixion and Resurrection, the Believer at that time is RE-Generated unto Newness of Life through the Indwelling of the Holy Spirit.
I agree with Adoption, and that can be seen in Galatians 4:1-7. However, I cannot see "baptism" or "believers baptism" referenced in that passage. That is a connection to you and your perceptions, but it is not in the text itself.
However, lets carry this allegorical implication forward and see what else Galatians 4 says about Adoption:
Gal 4:22-28 NASB
22 For it is written that Abraham had two sons, one by the bondwoman and one by the free woman. 23 But the son by the bondwoman was born according to the flesh, and the son by the free woman through the promise. 24 This is allegorically speaking, for these [women] are two covenants: one [proceeding] from Mount Sinai bearing children who are to be slaves; she is Hagar. 25 Now this Hagar is Mount Sinai in Arabia and corresponds to the present Jerusalem, for she is in slavery with her children. 26 But the Jerusalem above is free; she is our mother. 27 For it is written, "REJOICE, BARREN WOMAN WHO DOES NOT BEAR; BREAK FORTH AND SHOUT, YOU WHO ARE NOT IN LABOR; FOR MORE NUMEROUS ARE THE CHILDREN OF THE DESOLATE THAN OF THE ONE WHO HAS A HUSBAND." 28 And you brethren, like Isaac, are children of promise.
So the "adopted" children (the children of the desolate mother) are represented in the Abrahamic covenant by the birth of Isaac. What say the scriptures regarding righteousness:
Rom 4:8-13 NASB
8 "BLESSED IS THE MAN WHOSE SIN THE LORD WILL NOT TAKE INTO ACCOUNT." 9 Is this blessing then on the circumcised, or on the uncircumcised also? For we say, "FAITH WAS CREDITED TO ABRAHAM AS RIGHTEOUSNESS." 10 How then was it credited? While he was circumcised, or uncircumcised? Not while circumcised, but while uncircumcised; 11 and he received the sign of circumcision, a seal of the righteousness of the faith which he had while uncircumcised, so that he might be the father of all who believe without being circumcised, that righteousness might be credited to them, 12 and the father of circumcision to those who not only are of the circumcision, but who also follow in the steps of the faith of our father Abraham which he had while uncircumcised. 13 For the promise to Abraham or to his descendants that he would be heir of the world was not through the Law, but through the righteousness of faith.
So we see here that Abraham was credited as righteous before being circumcised. This happened a year before Isaac was born. I will shortly present passages showing that 'circumcision' represents the indwelling of the Holy Spirit, but what you need to realize at this point is that Abraham was "saved" or "made righteous" or "brought to life" prior to circumcision and prior to adoption (represented by the birth of Isaac).
Eph 1:13-14 NASB
13 In Him, you also, after listening to the message of truth, the gospel of your salvation--having also believed, you were sealed in Him with the Holy Spirit of promise, 14 who is given as a pledge of our inheritance, with a view to the redemption of [God's own] possession, to the praise of His glory.
Col 2:11 NASB
11 and in Him you were also circumcised with a circumcision made without hands, in the removal of the body of the flesh by the circumcision of Christ;
Deu 30:6 NASB
6 "Moreover the LORD your God will circumcise your heart and the heart of your descendants, to love the LORD your God with all your heart and with all your soul, so that you may live.
Rom 2:28-29 NASB
28 For he is not a Jew who is one outwardly, nor is circumcision that which is outward in the flesh. 29 But he is a Jew who is one inwardly; and circumcision is that which is of the heart, by the Spirit, not by the letter; and his praise is not from men, but from God.
So the indwelling of the Holy Spirit is "circumcision of the heart" and is foreshadowed by circumcision of the flesh in the Abrahamic covenant of faith. So when Romans 4 says that Abraham was made righteous while uncircumcised, we can infer then that in the covenant of faith, faith precedes indwelling and adoption.
We know, when anyone in the New Testament speaks of (The New Birth or Re-birth) they are speaking of The Spiritual Re-Birth because Nicodemus asked Jesus a specific question concerning this .
Joh 3:3-6
3) Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God.
4) Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother's womb, and be born?
5) Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.
6) That which is born of the flesh is flesh; and that which is born of the Spirit is spirit.
This then is the description on how the the Holy Spirit indwells the Believer through Baptism, having been Born Again we have been RE-Generated.
Therefore The Regeneration cannot be misunderstood to mean anything other than The Spiritual Re-Birth, and never to mistaken or assumed to be, a Physical Re-Birth because that is a Physical impossibility.
Mat 19:28 NASB
28 And Jesus said to them, "Truly I say to you, that you who have followed Me, in the regeneration when the Son of Man will sit on His glorious throne, you also shall sit upon twelve thrones, judging the twelve tribes of Israel.
Oddly, one of the regeneration passages contradicts your statements. Spirit's that have no bodies have no need or ability to sit. Jesus on his Throne judging the 12 tribes comes post-physical resurrection... so in this case I would surmise that the regeneration talked about in this passage must be physical.
1Co 15:35-37, 42, 44, 51-52 NASB
35 But someone will say, "How are the dead raised? And with what kind of body do they come?" 36 You fool! That which you sow does not come to life unless it dies; 37 and that which you sow, you do not sow the body which is to be, but a bare grain, perhaps of wheat or of something else. ... 42 So also is the resurrection of the dead. It is sown a perishable [body,] it is raised an imperishable [body;] ... 44 it is sown a natural body, it is raised a spiritual body. If there is a natural body, there is also a spiritual [body.] ... 51 Behold, I tell you a mystery; we will not all sleep, but we will all be changed, 52 in a moment, in the twinkling of an eye, at the last trumpet; for the trumpet will sound, and the dead will be raised imperishable, and we will be changed.
So like a seed goes into the ground and comes out of the ground in a different form (as a plant), a person's body goes into death in one form and comes out of death in another form. It is not that we will not have a body... we will have a body, but it will be a transformed or changed body. The Mat 19:28 passage is post-trumpet, so we will already have our new, immortal bodies. Like Ephesians 1:13-14 says, the Holy Spirit in us is a PLEDGE of the full inheritance (adopted children inherit)... not the full inheritance.