I've seen it 3 ways. Like you described, with foil over the grate, or in a pouch like you'd do vegetables. My dad always did the latter I just do it on the grill
No — are you prepping your grates? Preheat, clean thoroughly with a wire brush, then dunk a crumpled up paper towel in vegetable oil, hold it with your BBQ tongs, and brush the grate.
Sticking can also be a sign that the food isn’t cooked yet, it tends to release once it is seared so don’t be afraid to leave it alone for a bit.
The only time I’ve used foil on the grates before was when I was at a lake and the grill was filthy. I wasn’t able to adequately burn off all the gunk so went the foil route. There’s no reason to do this if it’s your own grill and you maintain it.