When the Gospel is preached by those who are sent, it is the Spirit; the Spirit of grace that is released to empower those who hear to believe and therefore obey the Gospel, thus being born again.
Here's what the Bible tells us about being born again:
[1Pe 1:21 KJV]
21
Who by him do believe in God, that raised him up from the dead, and gave him glory; that your faith and hope might be in God.
[1Pe 1:23 KJV]
23
Being born again,
not of corruptible seed, but of incorruptible, by the word of God, which
liveth and abideth for ever.
Above, you say becoming born again is from our believing and obeying the Gospel.
Instead, the verse above informs us of how we become born again. It says nothing of any requirement or attribute the recipient must manifest to become born again, to include either believing or obeying - although after the fact we will exhibit those traits.
Instead, it informs us that God does it all.
One of Christ's names as described in the Bible is the Word of God. So, to be born again, is to have Christ (God) indwell the recipient. In fact, the indwelling of God, at its essence is what becoming born again is (by the incorruptible seed). Subsequent Christ like actions of the true Christian will emanate from, and will be because of the indwelling.
[Rev 19:11-13 KJV]
11 And I saw heaven opened, and behold a white horse; and he that sat upon him [was] called Faithful and True, and in righteousness he doth judge and make war.
12 His eyes [were] as a flame of fire, and on his head [were] many crowns; and he had a name written, that no man knew, but he himself.
13 And he [was] clothed with a vesture dipped in blood: and
his name is called The Word of God.