I'm pretty sure that's why they're called cold sores, and I've had them happen during flus before, although not during the vast majority of them. Nevertheless, I haven't ever had a situation like I got with covid, and the actual covid symptoms were minor (after the first day, which was a weird intense generalized weakness that eventually went away.)
Aren't most variants of cold and flu members of the corona-family of virus? If so, it would seem logical that COVID also being a coronavirus would have the same (or at least extremely similar) effect on cold sores as other members of the same viral "family" would cause.
Covid is a cold, anyway. A strong, strange cold.