Both Jesus and Paul delegated that task to others, just as Peter had others baptize Cornelius in Acts 10. So you're logic doesn't hold up. It was getting done in every case
I reject your idea that being born of water has to do with physical birth
of course ot has nothing to do with physical birth. John had just been preaching the kingdom is near and baptizing for repentance and remission of sins in water . Which everyone in the region had come to partake of
“And there went out unto him all the land of Judæa, and they of Jerusalem, and were all baptized of him in the river of Jordan, confessing their sins.”
Mark 1:5 KJV
Then John said this
“I indeed baptize you with water unto repentance: but he that cometh after me is mightier than I, whose shoes I am not worthy to bear: he shall baptize you with the Holy Ghost, and with fire:”
Matthew 3:11 KJV
Then Jesus ministry begins immediately after johns
“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.
John 3:6
Look at Jesus baptism he was baptized in water and received the spirit .
“And it came to pass in those days, that Jesus came from Nazareth of Galilee, and was baptized of John in Jordan. And straightway coming up out of the water, ( image of birth coming out of the womb ) he saw the heavens opened, and the Spirit ( receives the spirit ) like a dove descending upon him:”
Mark 1:9-10 KJV
“Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, ( water )
and ye shall receive the gift of the Holy Ghost.”( spirit )
Acts 2:38 KJV
even happens in the opposite order it both water and spirit are always present
“While Peter yet spake these words, the Holy Ghost fell on all them which heard the word. ( spirit ) For they heard them speak with tongues, and magnify God. Then answered Peter, Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we? And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days.”
Acts 10:44, 46-48 KJV
Baptism has never changed but what changed is now we can receive the spirit. Through the name of Jesus Christ.
Should be as simple as this
“Then Philip opened his mouth, and began at the same scripture, and preached unto him Jesus.
And as they went on their way, they came unto a certain water: and the eunuch said, See, here is water; what doth hinder me to be baptized?
And Philip said, If thou believest with all thine heart, thou mayest. And he answered and said, I believe that Jesus Christ is the Son of God.
And he commanded the chariot to stand still: and they went down both into the water, both Philip and the eunuch; and he baptized him.”
Acts 8:35-38 KJV
when you enter into the water your being buried together with Jesus and when you emerge from the water it’s the imagery of being born anew as a child of God in Jesus his sons name receiving his spirit because of the promise to believers and remission of sins by acting in faith according to his promise “ repent and be baptized in Jesus name for remission of sins “

