This was a question that the pagan opponents of Christianity took quite seriously; in fact, they understood the question to be rhetorical. Jesus obviously could not be a son of God if he was a mere TEKTŌN. The pagan critic Celsus particularly mocked Christians on this point, tying the claim that Jesus was a "woodworker" into the fact that he was crucified (on a stake of wood) and the Christian belief in the "tree of life."
And everywhere they speak in their writings of the tree of life... I imagine because their master was nailed to a cross and was a carpenter by trade. So that if he happened to be thrown off a cliff or pushed into a pit or suffocated by strangling, or if he had been a cobbler or stonemason or blacksmith, there would have been a cliff of life above the heavens, or a pit of resurrection, or a rope of immortality, or a blessed stone, or an iron of love, or a holy hide of leather.
Bart Ehrman, Misquoting Jesus