I heard so often that it's outright dangerous to run when you're sick. Does anyone with knowledge know whether this is true? The general vibes of this thread gives me the feelings that it's more like a inconvenience than straight up dangerous to your heart and everything, which is what I heard and believed up until now.

Depends how sick, I have frequently heard that with mild head cold, that has not gone to lungs it is good to run to activate immune system etc.