I always put BBQ sauce on my chicken after it is done baking, but I was wondering if I can do the same while seasoning it to save time?
Yes you can, and should try it.
Baking the chicken with the sauce will cook the sauce and will start caramelize some of the sauce making the chicken better (IMO).
Yes, "It is possible"...however there are trade offs. The sauce will caramelize and become a tasty glaze...but...your chicken becomes more 'braised'(cooked in liquid) than 'grilled'. You get the caramelized sauce but loose the 'maillard effect' that comes from grilling. As taste is personal matter the best advise is try it and see which you and your audience prefer.
On an aside, you might try doing a brine (long soak) or marinade (other than the viscous BBQ sauce, for instance soy sauce) and/or even injections to layer different flavors.
Yes. Advantage is moister chicken, more tender chicken with a baked glaze on it. Seals the skin it will. I often rub a chicken inside & out with a wild honey& BBQ dry mix. Yard hens. If over baked at most all you need remove is the skin to discard.
Yeah, people make baby back ribs this way by gradually adding layers of BBQ sauce on the meat. You'll eventually get a good layer of sauce coating the meat.