Agree very much with this entire post, except one slight nit:
3570k will have significant performance benefit over a 8350, and by extension a 4770k will also be significantly better than a 8350. But yes, streaming on medium, probably won't stress enough to warrant a 4770 vs 3570. If you're gaming and have the budget for the parts you listed, please do not waste it on a subar AMD processor
If you are going to be doing a lot of rendering, you can take advantage of 8 threads vs 4. Depends on how much you value your time, and exactly what you're doing. Some games require more CPU (BF4 multiplayer may be one of these, go look around for benchmarks to check). If your game is actually going to use 4 cores, the SMT from a 4770 may actually help quite a bit while streaming.
Streaming will also specifically depend on your internet. If you have shit internet, don't bother spending a lot of money on CPU, because you won't be able to push out a great stream anyways