I built a system recently for almost the exact budget and use case. I went with a 10105F (6k) and a used GTX 970 from the marketplace here for 7k. There were a couple of 7-8k cards posted recently I think, and they do seem to come up now and then.
Anyway, while the 5600G isn't bad IMO, my total spent of 13k on discrete CPU+GPU, even if both a couple of gens older, still would give me better performance in my book,