I don’t know why the suggestion is controversial.
My experience is that above the neck the risk is spreading the infection. Below the neck interferes with breathing which is fundamental to any aerobic activity.
What I do, which may not be best practice, is for a head cold just run easy as long as I am not fatigued; and for anything below the neck stop until I think running will help break up the congestion. It is important to keep in mind that when you are carrying fatigue you are also more susceptible to infections.
There seems to be a point where resuming training accelerates recovery. If you hit it it helps, but I also have missed it and prolonged the infection for weeks. Worst case, I once ended up with walking pneumonia for months…