Like I said, it doesn't have a correlation with the times that silver was regularly used in mirror production, nor where such would be regularly used and where you'd find the vampire myths popping up with mirrors.
If you find evidence to the contrary I'm happy to be corrected, but I did a bit of research into it when my wife's TikTok popped up with that "interesting fact" and it seemed like bunk.
Silver mirror | Roman | Early Imperial | The Metropolitan Museum of Art
Richter, Gisela M. A. 1915. <i>Greek, Etruscan and Roman Bronzes</i>. p. 288, New York: Gilliss Press.<br/><br/>Oliver, Andrew Jr. 1977. <i>Silver for the Gods : 800 Years of Greek and Roman Silver</i>

1st century silver hand mirror from Rome. These things were heirlooms and expensive as all heck. They were also something that continued to be made for particularly wealthy people all the way into the 1900s.
Silver mirror | Roman | Imperial | The Metropolitan Museum of Art
Milleker, Elizabeth J. 1992. "Ancient Art: Gifts from The Norbert Schimmel Collection: Greek and Roman." <i>Bulletin of The Metropolitan Museum of Art</i>, 49(4): p. 62.<br/><br/>von Bothmer, Dietrich, Carlos A

The horizontally handled Roman mirror died out around 900CE. This one's also from the 1st century.
So, y'know... Silver Mirrors have continued being a thing for 2,000 years. And like white wedding dresses they were a sign of wealth and privilege, something that gets filtered down into the populace even if it's not kept true to its roots.
As far as silver being associated with purity, that goes back to the various deities of the moon and virginal goddesses it's been associated with for a few thousand years. But it also isn't just werewolves silver is supposed to ward off or contain. Things like mirror-lined or silver-lined boxes are meant to contain evil spirits and keep them from escaping, for example.
And also of note: It doesn't matter WHEN the myth about Vampires, Mirrors, and Silver came about. Whether it's witches or Bram Stoker, the vampires we deal with in pop culture are not the Vrykodlak of ancient myth. If it works for the story and makes sense to the primitive part of our brains that go "Oh that explains it" then the magic works, you know?
Vampires didn't command beasts or turn into bats or become mist in the ancient myths. They were wandering corpses that would slowly drain your life away and retreat back to their grave before dawn. Grave. Not sumptuous velvet lined coffin with a handful of dirt spread across the bottom of it. And they certainly couldn't make Vampiric Thralls back then, either.
Of course give it another generation or three and maybe they'll sparkle in sunlight!
Call it a "Sparkling Interpretation" of vampire myth since it doesn't come from the Vampire Myth part of France, if you like.