I hesitate wading into this discussion because of the egos involved, but what the hell, apparently my ego is huge too.
I will occasionally throw a rope down a pit. Usually short pits (50' or less give or take) and usually where I can see the rope going down as it's falling. I also usually throw the first 20-30' of the rope down most pits I'm rigging to get enough weight to readily lower the rest, and in both instances I am holding the rest of the rope so I can get a feel for what it's doing. Lowering gives me a good feel for what's going on with the rope as it's deploying, and I've occasionally avoided problems because of this method.
I never throw the entire rope down larger pits, and I don't care to be the first person on the rope when someone has. I also teach my students to lower ropes down drops. I sort of look at throwing the entire rope as one of those Hollywood things, it makes for a great visual but really serves very little purpose. Yes, there's rapid deployment of the rope, but I will counter that with the potential for spending more time dealing with tangled rope and the distant possibility of rope damage. I won't claim that damage is something I worry about, but I can't also claim it doesn't happen as I've seen it too.
I've also occasionally rappelled on the rope coming out of the bag, but I've now been on two rescues because of this technique, one from someone short roping the pit without a stopper knot, and once because a mouse had made a nest in the bag and chewed through the rope (in many places). About the only time I will rappel out of the bag is if I had stuffed it recently.
So y'all do what you want because you will anyway