You assume that a need existed to elaborate on the basic teachings. At the same time it is unlikely that they were trying to spend too much time explaining things to deniers of Christ.They wouldn't need to copy John 1 into every letter they write but it just seems that they would have further elaborated, expounded on such an important doctrine which since according to some, the Trinity has to be believed in order to secure salvation.
I don't believe John 1 says that Jesus is God . . . it may be a so-called doctrine of inference but as far as clear, explicit statements stating a clear concise Triune perception of God - there are none.
Why would he be called 'the Son of God' because he is/was the Son of God . . .
"And behold, you will conceive in your womb and bear a son, and you shall call his name Jesus. He will be great and will be called the Son of the Most High. And the Lord God will give to him the throne of his father David, and he will reign over the house of Jacob forever, and of his kingdom there will be no end.” . . . . And the angel answered her, “The Holy Spirit will come upon you, and the power of the Most High will overshadow you; therefore the child to be born will be called holy—the Son of God......"
Why would that result in "God incarnate"? God caused Mary to conceive . . . God wasn't the one conceived . . . His Son was conceived. God is not HIS SON - God is his God and Father.
Then, do you really doubt that Christ could not the Son of God? That is in violation of the testimony of scripture. Jesus is called the Son of God because of being divine with his Father while becoming incarnate through Mary. Jesus does go not speaking of himself Son of Mary but rather as Son of God. There is no other reason I have heard for him to have that unique status and distinction.