It's situational, and that depends on what kind of Monitor setup you are using. If you're gaming on a 144hz monitor then you will want the fastest GPU and CPU you can afford. Once the FPS reach over 100fps then the processor can bottleneck so obviously you want the fastest CPU available. The i7 does benefit gaming but it's usually because of the higher clock speed and there are some games that use the Hyperthreading.... but the only time you will see the benefit of the i7 is with a 144hz monitor. That's the catch behind the mighty Core i7.
Back in 2011, there was no difference between the Core i5 and Core i7 in gaming because of the way games were programmed at that time. However, the last couple of years these quad-core versions of the Core i5 and Core i7, the Core i7 was really starting to pull ahead in some games. Now that the 8th Generation chips have 6 cores, the difference is 1% to 3%.
But if gaming on a 144hz display is your goal then a Core i5-8600k and the GTX 1080ti would be a better than the Core i7-8700k and the GTX 1080. There is a bigger difference between the 1080 and 1080ti as opposed to the 8700k and the 8600k.
If you're going to game on a 4k display that runs at 60hz then a $200 CPU will do. You could actually use a Core i5-8400 or a Ryzen 5 1600x and you would be fine. If you plan to buy a 4k monitor then the Core i5-8600k and GTX 1080ti is Hands Down the better pick.
IMHO, if you're going to stream then it's just a better bet to do the streaming from a 2nd PC with a capture card. The Core i7-8700k would be much better for video editing and rendering since those are thread heavy tasks.