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This question is a bit tricky. Mundane every day facts like descriptions of writing tools tend to be poorly documented because most people find them uninteresting.
However, first to disabuse you of the idea that lead can't write. Have you ever handled lead yourself? It is remarkably soft and malleable, and easily makes marks on hard surfaces, although not nearly as easily as graphite. I'd encourage you to find some yourself, except it might be tricky - you may need to look for a tradesman, a scrap yard, although it might be illegal where you live... plus it is mildly poisonous. Instead I'll just point to the existence of metalpoint, a traditional drawing technique where a soft metal like silver or lead is dragged across an abrasive surface:
Metalpoint is a drawing and writing medium that dates from antiquity and was particularly popular from the 14th century to the beginning of the 16th . The technique of metalpoint involves dragging a stylus of metal across a substrate prepared with a slightly abrasive surface coating, or ground. As the metal is drawn along the surface, tiny particles of metal are left behind, creating a mark.
Metalpoint Drawing: The History and Care of a Forgotten Art
Beth Antoine
Albrecht Durer did many drawings in silverpoint, such as this one:
Albrecht DΓΌrer [Public domain], via Wikimedia Commons
As for whether lead styluses were used in ancient Rome, to write in this form, is a bit uncertain:
The lead styluses used in ancient Rome were actually used to write on waxed tablets - the stylus would create engravings in the wax, as opposed to breaking off and leaving a mark. It looked something like this:
Photo by Peter van der Sluijs CC BY-SA 3.0
The reason why the styluses were made of lead was because the ancient Romans used lead everywhere - it was cheap, plentiful and easy to use.
So did ancient Romans use leaden stylus-shaped instruments to write, in the manner of leaving lead marks? Unfortunately I couldn't find good sources, but there are these:
This page claims that ancient Romans (also Egyptians and Greeks) used lead discs to mark papyrus:
The ancient Egyptians, and the Greeks and Romans, too, used a small lead disc for ruling guide line on the papyrus to keep the lettering even. The Romans called it a plumbum - Latin for lead.
So from here it's not a terribly big leap in logic to assume that stylus-shaped lead was also used for writing occasionally.
The first documented description of wood-cased lead pencils comes from Konrad Gesner in 1565:
The stylus shown below is made for writing, from a sort of lead (which I have heard the English call antimony), shaved to a point and inserted into a wooden handle.
I haven't found the primary source, but assuming that's an accurate excerpt, it should suggest that wood-cased lead pencils existed at least in the 16th century. Don't be distracted by the coincidentally similar date to the discovery of graphite - it took quite a while for the usefulness of graphite at writing, let alone in wood-cased form, to become widespread. In fact, encasing graphite in wood came much much later, during the 18th-19th centuries. Early graphite pencils were wrapped in various materials like paper, twigs or string.
So was lead used to write in ancient Rome? Most likely. Did they use lead pencils? It's uncertain, but not unreasonable.