Well, dying a cloth is a very different process than puting ink on top of the surface of a parchment, so the rarity of black dye doesn't apply to inks.

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You're right that most medieval inks where metallo-gallic ink. Those were made not only with oak galls, but mixed with iron sulfate. It is important to understand that it is not a pigment in the gall than produce the black, but a chemical reaction that makes it appear when the tanin of the gall and the iron sulfate are mixed and heated up. A cloth dye usually sticks to the fabric thanks to the addition of aluminium salts; but for an ink, you will rather add som gum arabic or other oily stuff, so that it will stays on top of the parchment rather than bleeding inside it. I have tried it myself (from a 12th c. recipe by Theophilyus Presbyter), it is quite easy and produces an ink of surprislingly good quality, and yes, a very black one.

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So now to your last question: as I told, the result at first is indeed a very black ink. But as that it contains iron (remember, "metallo-gallic"), this iron can slowly oxydate in the years/centuries after. So yes, in some case the ink has faded over time, but rather to a slightly rust-brown color.

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A last point: the metallo gallic ink was not the only kind in use in the middle ages. The carbon ink (like the modern "india ink"), made from burned wood smoke and gum arabic, was also in use and well known since the Romans. It can also produce a very good black.

u/sunagainstgold partially answered your question, but touches on fabric dyes more than inks.

It's worth a read though and I hope you find this helpful. Following to see if anyone can more directly answer the question.