Actually, lack of shavenness is scriptural. Clean shavenness is not.
I watched on the Satanic TV (History/Discovery) not so long ago a "reconstruction" of what Jesus might have looked like! They dug up a skull of a man from that area and era, and showed how the Jewish skulls differed dimensionally from Gentile skulls (funny, that's what Hitler was trying to prove!), being more oblong or round than the skull of a non-Jew. After some computer engineering graphics trials, they came up with this olive-skinned Middle-Eastern looking character they said was "probably" what Jesus looked like.
Of course, they planted on him short, somewhat curly hair, and a short beard. The truth is, even +Paul says that man should have short hair, which seems to contradict the OT commandment, but the truth is that the Jews had styled hair and beards. Pious Jews didn't. Any kind of styling is seen in Jewish and Christian humility as being vain, drawing attention to one-self, and is to be avoided.
Thus, Orthodox icons always depict Christ with unstyled beard and long hair, which is what you will see in Orthodox clergy coming from Orthodox cultures -- Russia, Serbia, Greece, etc. In America, Orthodox priests are either clean shaven or have styled (clipped) decorative beards which is completely out of character with the Orthodox mindset and 2,000-year-old practice.
Which only shows that everyone picks and chooses to his liking and that egos play a much more prominent roles in our lives than the illusory image we have of ourselves would allows us to admit.
I saw that program. I'm not surprised to learn that they showed Him as being an impious Jew. I also remember asking myself: "what are they saying, that all Jews look alike"? :)
As I think of it, I am reminded of the depiction of Jesus and the disciples in "The Passion". IIRC, they didn't have below-the-shoulder length hair. If that's right, would that have been a mistake? Assuming you saw the movie, did any other mistakes jump out at you?