Hi, u/Aroused_AI, sorry for this very late answer. It was already late when I found it and it took me some days of research, but nonetheless I desired to give it an answer. So I hope you don't mind a 25 days belated answer and I hope as well that you appreciate it. Okay, this is going to be an obnoxiously long wall of text, so let's get started.

Your question is about pest control in the Middle Ages, but it's founded primarily on the assumption that in the Middle Ages cats were massacred en masse causing a shortage of rodent-killers, and this assumption is founded on what is written in the article you linked (and all over the Internet tbh). So, firstly I am going to analyse if it's reliable (spoiler: for the very most part, nope).

The author of the article, a certain Mr. Mark, is not an historian, so I'll take what he says with a barrel of salt. He quotes the words of only two people: Desmond Morris and Virginia C. Holmgren. Morris is a famous biologist, ethologist, zoologist, author of the famous book The Naked Ape; the book referenced here is Catlore (1996). Scientists aren't exactly famous for being good at history, and examples of this abound. Virginia Holmgren, I admit, was unkown to me until now, but anyway she's not an historian, just a writer. I'd say that theirs are the only books used by Mr. Mark, and the others were thrown in the bibliography just to make the article seem more academic. It's a bold assumption, but the inclusion of Susan W. Bauer's History of the Medieval World (which despite the title covers only Late Antiquity and Early Middle Ages, periods not dealt with by the article) and other blunders we'll see later make me feel quite confident about it.

The first quote from Morris in which the two words highlighted in red (?) are "Christianity" and "religion" makes me think there is a certain anti-Christian stance in the article. Of course, that's not a problem per se: a prejudice becomes a problem when it prevents you from getting history (or anything else) right. I don't know if Phoenicians really spread the association of cat with witchcraft, since I'm no expert on Caananite peoples and their religion, but what follows somehow makes me sceptical of it.

Hecates, cats and Liberalis?

Hecates was, as is described in the article, the Greek goddess of magic, sorcery, ghosts and the underworld. When the Hellenistic world saw an exponential growth of superstition (I century BC and on), her cult grew very common that rituals and divination became a pretty important part of the mysticist strain of Iamblichean Neoplatonism. Divination with Hecate's strophalos (or iunx, as a Byzantine erudite called it centuries later) was practicised even in Athen's Neoplatonic school. The closing of this school and the prohibition to play with this kind of fidget spinner hindered scientific progress for at least three millennia. Now, the article states correctly that Hecates was associated with dogs, but it doesn't explicitly explain why: she was a liminal goddess, staying between the reign of life and that of ghosts. The Dog, with his function of guardian, was her animal-symbol. People sacrificed to her dogs, impaled them next to her shrines built in the middle of crossroads (another symbol of liminality). These sacrifices are attested by multiple sources, so it's certain that they happened, probably on daily basis. Sources that attest the consummation of these sacrificed dogs are more scant, but since all sacrificed animal were usually eaten, it's not unlikely. The article then narrates in a very confusing, inaccurate and misleading way the myth of Heracle's birth and states that it further linked the cat with witchcraft as Hecate took the now-turned-cat Galinthius [sic] in her reign with her. The fact that Mr. Mark states that this myth was "given greater boost by the Latin writer Antoninus Liberalis", is to put it mildly curious, as Liberalis' version is the only one in which Hecates even appears and the only one in which the animal Galinthius [sic] was transformed into, was interpreted by some authors to be a cat. Now, after a fast googling articles with Galinthius-Hecates-cats-massacresofcats-plague easily crop up like shrooms, so I feel like it deserves a debunking even if it's tangential to your question. Firstly, it's Galanthis or Galinthias, not Galinthius, as it's written thrice in the article. Galanthus is a genus of flowering herboceous plants; since in those articles all around the web I've come across this spelling, the culprit here is "copypaste any pseudohistorical gargabe you see without fact-checking". Then, Antoninus Liberalis was not a Latin writer, he was a Greek writer. We don't know anything about him; maybe he had the Roman citizenship, but however he almost certainly lived in the Greek part of the empire. I say this because his Metamorphoseon synagoge arrived to us in a Byzantine manuscript, while in Western Europe he would've been totally unknown till the fifteenth c. - contra what the article says. The statement that Liberalis' version gave a greater boost to the association of cats with witchcraft is without evidence, and being Liberalis' version the only one with Hecate and, maybe, Galinthias-turned-cat, it cannot have given greater boost to another previously non-existent story with Hecate and Galinthias-turned-cat. I think it's clear. Anyway, the myth of Galinthias' metamorphosis that was really famous (and more importantly remained famous in the West after 500 AD to literate people ie clergymen), was that spendidly narrated by Ovid in his mythological work Metamorphosis, Book Nine, 273-323. In this version of Hercules' birth, her mother Alcmene is having great difficulties to deliver the child, so asks Lucina (goddess of childbirth) to help her. But Lucina has been ordered by Juno/Hera to prevent the delivery and make both mother and child die in the throes of labour. But Galanthis "flava coma" manages to deceive Lucina so that Alcmene can deliver his child Hercules. Hera, in a fit of rage, transforms Galanthis in a weasel (in Liberalis' version she is transformed by the Moirai/Parcae.

Cont. below