I carry my throwable in a rear tramp pocket, with the throw strap exposed and easy to grab. When I didn't have a tramp pocket, I used a bungie over the cushion to the tramp lacing. I've always tried to keep them ready to deploy.
I've used it at least once. One time I came upon a capsized small row boat or canoe or kayak. Two adolescent girls were fumbling around without any flotation available to them except the boat. I tossed the cushion, then came back around and tossed two PFD and instructed them to put them on.
As far as a line attached, there are differing schools of thought. My thought is that you may be going so fast that if the throwable is attached to the boat, it may be yanked away from the overboard person. Here is how I handle it:
I keep a 50' floating poly line available that I can tie to the cushion if I want to. When I go on my serious Channel Island Cruises, I take a 50' or 100' (can't recall) floating poly line and tie it off to the boat on one end. On the other end I tie a 500ml pop bottle to it and fill it half way up with water. This can be thrown and it will float.
I tie the rope into one continuous "daisy chain." and shove it all into my rear tramp pocket.
My plan: Toss a throw cushion toward the person in the water and then make a turn to come back. When close enough, toss the water bottle to them.
GARY