Your argument hinges on a denial that John 3:5 refers to baptism. If John 3:5 refers to baptism, then that is, itself, the Bible connecting baptism with birth.
Further, the language of resurrection, namely our being raised with Christ to new life is itself the language of new birth.
"We were buried therefore with Him by baptism into death, in order that, just as Christ was raised from the dead by the glory of the Father, we too might walk in newness of life." - Romans 6:4
In baptism we are buried with Christ, therefore having died with Him, and therefore even as Christ was raised, so we now have new life. That new life, that is our being a "new creation" (2 Corinthians 5:17), that is regeneration, new birth, born again.
See also Titus 3:5,
"He saved us, not because of works done by us in righteousness, but according to His own mercy, by the washing of regeneration and renewal of the Holy Spirit,"
Resurrection means new life, new creation, it means regeneration for us. Christ has been raised, and we--united to Him--are alive with Him, we share in His life, by the Holy Spirit, and that makes us new people, born again.
So, for one, Scripture does connect baptism with rebirth, it does so in John 3:5; but even if one wants to deny this explicit mention in Scripture, it is hard to ignore the larger consensus and witness of the Bible: Christ has been raised, His life is granted to us that we might share in it, and thus we have a new life, we are born again, alive to God by His grace and Spirit. The language of resurrection is the language of newness of life, the language or renewal, new creation, rebirth, etc.
On top of all this, one would then also need to ignore the unanimous and universal consensus of every Christian from the time of the New Testament until early modern times, where baptism and regeneration have always been connected based upon this very text from John 3:3-5. The argument then is literally every Christian writer, before the advent of the Anabaptist and Baptist traditions, has been wrong. And while one may very well believe that, that's a pretty difficult sell.
And if you want to content against that assessment, then I would be very curious if anyone can provide any ancient Christian sources which don't understand John 3:5 as being in connection with baptism.
-CryptoLutheran