One recommendation is to hold down on the GPU for now as GTX1180 is gonna launch soon. On paper it is 50% more powerful than GTX1080 which is a huge improvement. I strongly recommend waiting for it. Get the PC but hold down on GPU.
Also, what would budget have to be raised to in order to justify a i7-8700K and Z370 chipset motherboard? Part of me wants him to wait for i7-9700k, but I don't know the release date.
Intel i7-9700K of Ice lake series will release in 2019. That is a long wait. NVIDIA Turing is launching in few months I recommend waiting for instead and not having expectations on launch of Ice lake any soon.
This PC will definitely perform better for gaming than the Ryzen build. But can loose bit of gaming performance while steaming compared to Ryzen build. Ryzen build is more consistent with Fps while Streaming. Intel build is better in gaming performance.
If he games most of time and rarely Streams it go for Intel. But if he regularly Streams go for Ryzen.