Which would raise the question of Divinity vs. paranormal activities linked to a (most probably) recessive gene. But miraculous intervention has a long history in Judeo-Christian religion prior to the arrival of Jesus. Miracles were reportedly performed by Moses, Elijah and Elisha to name but three, none of whom made any claim to Divine status. Also, in either case, the descendants of Jesus would have to be remarkably fecund and widespread. Reports of miracles on every continent and subcontinent and among virtually every racial grouping.
There wasn't really a concept of paranormal back then. People attributed supernatural things and events to divine figures and people acting on their behalf. Look at the swoon theory which attributes Jesus's resurrection to natural causes, which suggests that if you tried hard enough, you can explain every miracle to natural phenonema which people have exagerrated.
We don't need all miracles everywhere to be evidence of Jesus, we just need evidence of Jesus's bloodline being more favoured than others. Perhaps they are, since the Merovingians are according to one (almost certainly false) story descendents of Jesus. I think more monarchs would claim descent from Jesus (which incidentally also means claiming yourself as the Messiah, or at least his successor, not that the vast majority of Jews would accept you as such) if Gnostic-esque beliefs of Jesus being married were more accepted and widespread, and that takes an a 1st century POD (and possibly one in the lifetime of the historical Jesus if indeed Yeshua bar Yosef wasn't married).
The idea of a Jesus bloodline would fit very well into Greco-Roman religion with its many legends of offspring of unions of gods and mortals. It would fit so well it would detract from the idea of Jesus as the Messiah.