Why an earth would you get a Ryzen 1600 and go through the hassle of a total platform change when you can just drop a 7700 in there with zero hassle and also get better gaming performance.
Just get the 7700 or 7700K. It's not true to say there's no benefit to the 7700K if you can't overclock as the base and boost clocks are higher than the normal 7700.
You can use the 7700/K with the same cooler as comes with the G4560 though temps will probably run very hot (it's a hot CPU).
Might be better idea to get a 6700 or 6700K. They run much cooler and have very similar performance.
Also, a G4560 is a good CPU for e-sports titles but it's really bad for CPU intensive titles. So in something like BF1 online 64p you would go from drops to 25-30fps to a rock solid 60fps at all times.
An i7-7700 or i7-6700 (or K models) is a good match for a GTX1060 6GB or above.