From a Syriac description of the unicorn hunt:
They lead forth a young virgin, pure and chaste, to whom, when the animal sees her, he approaches, throwing himself upon her. Then the girl offers him her breasts, and the animal begins to suck the breasts of the maiden and to conduct himself familiarly with her. Then the girl, while sitting quietly, reaches forth her hand and grasps the horn on the animal’s brow, and at this point the huntsmen come up and take the beast and go away with him to the king. - Likewise the Lord Christ has raised up for us a horn of salvation in the midst of Jerusalem, in the house of God, by the intercession of the Mother of God, a virgin pure, chaste, full of mercy, immaculate, inviolate.
No wonder unicorn horns were so rare. To get them, you needed the help of a pure, chaste, immaculate virgin who didn't mind going to second base with a goat-horse in front of a bunch of hunters.
Also, the Wikipedia article on unicorns blithely states that they can always fly. With no reference, because this is common knowledge. Whereas anyone with half a brain knows that unicorns can always teleport--once per day, within the boundaries of its forest. (Uni was such a twink.)
So perhaps alicorns are rare because while it was easy enough to round up one curiously amiable virgin, a real unicorn hunt required two, one to catch the unicorn before it used its teleport and one to catch it afterward. That, and the unicorn had to fail its wisdom check, which is a real bastard at 21 WIS. Wouldn't manage it unless someone used a loaded die that rolled natural 20's, which was more feasible back in the Middle Ages, what with poor dice quality control, but more dangerous because in those days, DMs were armed.
More people died in the Middle Ages from failed attempts at munchkinning than from smallpox. It's true.
Anyways, you learn something new every day. I feel myself to be a deeper, richer person now, and I hope you do, too. Now if you'll excuse me, I feel the strangest urge to look up virgin porn.