Quite honestly it depends what level of games (graphically) you're streaming. I'm just saying with no GPU and a mere CPU with onboard graphics, your choices of what games you can stream become much more limited.
The other thing, the REAL elephant in the room, is it's just an odd build for game streaming. Below I embedded a video that demonstrates how even a mere $92 AMD 2200G APU can kick butt to the 630 onboard graphics of the 9600k.
The FPS you play at and the level of graphics settings you have to use in game has a LOT to do with how good a captured and uploaded video will look. Even more so with streaming live while playing, because for that, performance is even more of an issue.
Now if what you want to do is build what you can afford now and maybe add a GPU later, it would make sense to wait for the Ryzen 3000 series to see if it will have onboard graphics. None of the currently available Ryzen models have onboard graphics though, so that may be unlikely. A decent CPU is also much better balanced to a good discrete GPU, and they also have more Pci lanes than the APUs. That said, it's looking like the Ryzen 5 3600G may be a beast of an APU with a possible 8 core, 16 thread and perhaps 16 Pci lanes (possible release Q3 2019).