Arrive alive isn't exactly a good source. Occasionally they include some half-truths in their advice.
I researched it a bit because it piqued my interest:
It seems to boil down to the moth effect. You focus on something (oncoming car) meaning you're looking slightly right. If there aren't strong visual clues for driving straight (say road lines) then you tend to inadvertently steer slightly towards the direction of the object you're fixated on. Flashing lights do not seem to directly cause this, but could make it more likely that someone fixates on the object. Seems to affect motorbikes more than cars.
Source
I just sort-of evaluate each situation as it comes. I'll definitely consider emergency lights though - I recall being in a heavy downpour once were you just couldn't see the red lights, but you could see the flashing yellow. The effect was pretty dramatic tbh. Maybe fog is different - I haven't encountered even fog to judge it.
Mouse has a point though that it takes away the ability to warn about a *real* emergency, so one needs to balance the benefits. Personally I'd prefer if emergency lights are used for real emergencies and "oh sht I can't see anything" fog/rain.
One thing where I wished more drivers did use their emergencies is on GP highways to warn of traffic jams ahead. Sometimes you come around a bend at 120kmh and then there is this wall of cars standing dead still. There it is useful to be warned by drivers currently braking hard that there is trouble ahead. Hell some of the Audis etc have this programmed into them so they flash the emergencies on their own under hard braking.