The Slate article is not presented as serious scholarship, so I won't comment on it as such. (I do like the fact that the very American-holy-card portrait of an Aryan Jesus is captioned as "Photograph by ...")
The article leans toward "masculinity" as a reason for beards. Well, yeah, outside of circuses a Y chromosome is probably necessary, but a beard still seems the natural state of things, not some macho declaration. Or course I'm biased, but what's so natural about scraping your face with a sharp blade?
But religion does relate to beards. While the Qur'an doesn't mandate them, many Islamists feel they must emulate the Prophet. Orthodox Jews (haredim and hasidim and others) also grow beards or, as I've noted, don't shave.
One disagreement with the article: it says, after including the praises of beards by the early Christian Fathers:
The beard soon fell out of favor among clerics, though, and Christian holy men were forbidden to sport facial hair for several centuries before the ban was relaxed during the Renaissance.
This is, unsurprisingly, the usual Western tendency to equate "Christianity" with "Catholic and Protestant Christianity," (the link is to the Catholic Encyclopedia), though later they note that the Orthodox expect their clergy to be bearded.
Indeed they do. Coptic Pope Shenouda III is an example (above left), though the Ecumenical Patriarch of Constantinople, Bartholomew (below left), probably has him beat hands down. If you'll look at almost any Orthodox icons, or Coptic or other Eastern Christian saints, you'll see the modern prelates are only hoping to reach the glories of the patristic beards.
No huge moral here. I don't plan to shave, but I don't expect I'll let it grow to patriarchal lengths, either. This post is — as I assume the Slate post also is — a lighthearted reflection, not some ideological or theological declaration.