Additive...
In very precise product lighting, light is NOT always additive in every case, simply because the positioning of the lights can be such that light only reaches the bit it's aimed for.
Ecample: The first shot of the SuperFast head, lit with a strip softbox from below is positioned so that it only really lights the bottom bit. But, because the softbox is longer than the flash head body, some light also wandered into the inside of the reflector.
So, when the second light was added above the flash head, little if any of the light reached the area lit by the bottom one, so the light there wasn't really additive - but again light from the end of the softbox wandered into the inside of he reflector, which was then lit more brightly than before, so that light was additive. And when I then added a honeycombed light that specifically it the inside of the reflector, that light was entirely additive, it added to the light that was already there.
In your own example, my guess is that there was enough light from your hotshoe flashgun to lighten the shadows a bit, but not enough for it to blow out the part of the face lit directly by the sun, simply because the sun was so much brighter. It was still additive, but not enough additional light was added to make it obvious.
Personally I only use hotshoe flashguns for the jobs that they do well, but that's just me showing off because I have studio lights too. Hotshoe flashes are extremely limited when used when studio lights are a better choice, using a hotshoe flashgun instead of a studio light is a bit like saying "Well, I'd like to have a sports car, and I would like to have a Transit van too, so I'll buy an estate car, because it will do both jobs" when in fact it will sort of half do both jobs. If your hotshoe flashgun has either strobe or HSS then of course you can use it as a temporary modelling lamp, which is better than nothing, and if it doesn't then you can still use it, but there's going to be more trial and error.
My approach
is a bit down to earth, but that's the whole idea. There seem to be a lot of people out there who are either just showing off their dubious skills and who pretend that it needs a shedload of equipment and a lifetime of experience to light an egg or who produce videos that are basically just designed to sell overpriced lighting gear. I just try to show people how easy it really is.
Edit: Juggler, yes, that's another approach and a perfectly valid one. But although I do use that technique myself sometimes, it's a hard work way of doing it, and I generally go for methods that are far less prone to producing lens flare. And I'm not trying to sell overpriced lighting gear
