The i5 wipes the floor with the Ryzen, especially since it's easily OC at 5Ghz if you get a decent chip(it's also cheaper or at the same price usually). Stop trying to push other gear down his throat. In gaming the i5 7600k is a very good CPU and no it will not bottleneck anything.
But also if your monitor has more than 60Hz you will have a small stutter on very demanding games at that resolution (if you have this issue set your Adapter setting to 60Hz). Besides those very demanding games you should run very smoothly on most games.
Also from a video I just saw, the 7600k isn't way better than the 2600k but for a good cooler you can probably OC it and get double the performance gap in games. (like 20 fps more instead of 10)