The VeryWell Health article doesn't side with the concept of "cold makes you sick" instead, it provides further context as to why many people would believe so. The main cause of it all is the increase in flu, but this doesn't mean the cold causes the illnesses. The article further bunks these misconceptions and provides further information on how to boost immunity. Based on further research, though, the website is a news site that has provided health information from health professionals for the sake of wellness being. This article specifically was written by Patty Weasler, who's a registered nurse with over a decade of experience, which was later reviewed by Jane Kim, who's board certified in Internal Medicine and also consults for physical medical education, so the article can be trustworthy for readers. If we'd like further proof, though, other news articles tend to say the same. For example, Heart's article, which is the American Heart Association, known for being the nation's largest and voluntary association, claims the same ideas. According to their article, "Viruses may survive and reproduce more easily in the cold, dry air of winter." further backing up the claims. So overall, we can assume that the claim "cold weather makes you sick" is not true.
sources: Verywell Health Article
Heart Article