From personal experience on bikes with oil coolers.....you will get some residual oil in the cooler which can take a while to drain out. I also run the engine before draining to warm it up, thin it out, then it drains better. Provided, of course, you can access the drain bolt easily. The joy of those old school 4-1 pipes that obscured the drain bolt.....
When refilling...I would do exactly as you planned. Run it....let the system find it's level....then top up when it's all settled.