My question is how the New Testament writers came to their belief in the Divine Sonship of Jesus of Nazareth, if it was not connected with the historical impact of Jesus himself.
The issue of the historical Jesus is a huge one, but a few facts can be our anchors.
First, the NT writings (most of them) can be reasonably and reliably dated in the lifetime of the eye-witnesses of the historical Jesus. Paul, whose letters are very close in time and in contact to the historical Jesus and his circle, who claims to be handing on the traditions he received (1 Cor. 15), is the strongest evidence, and his clear teaching about the divine identity of Jesus is undoubted. So we can date this testimony within 20 years or so from the life and resurrection of Jesus (see dating of Corinthians, early 50's CE).
Also, the canonical Gospels themselves can be dated in the first century, even allowing for the fourth Gospel to be placed at the close of the century , just within the eyewitness life-spans. There was a continuous community of testimony. As Richard Hays shows in his latest book, the Christology of the Gospels is the same in essence to the other NT writings.
Sceptical scholars can dispute the dating of NT writings but my guiding principle is to rest on sufficient evidence. We have some strong date indicators about the short interval between Jesus and the written belief in his Divine Sonship. The letter to the Hebrews, for example, was surely composed before the destruction of the Temple in 66-70 CE. The whole argument of the letter is the obsolete or fulfilled nature of the Jewish sacrificial system. Had the Temple already been destroyed, it is inconceivable that this fact would have been omitted by the writer; it would have clinched his argument.
So I give the benefit of the doubt to the Gospels when it comes to presenting reliable historical testimony about Jesus, recognising that there are theological themes governing the presentation. I see no reason to believe that the Gospel writers invented a legendary Jesus while so many living witnesses were alive.