Going to a newest generation CPU is going to be costly due to the fact that you’ll need both a new motherboard and DDR4 memory. The latter is very expensive right now. I’d say for your needs upgrading to a 4790k is likely the best bet.
The performance increase in games with a 7700k or 8700k will be fairly small as games are mostly GPU bound. Multi-core performance will be better with a newer chip supporting DDR4 like 7700k or 8700k, but may not be worth it for you since your main multi-threaded use is streaming games.
Here’s a great video highlighting the gaming performance of systems with the i7 chips from 3770k up to 7700k. Likely the performance difference is due to the inclusion of DDR4 for the most part, because the chips using both DDR3 or DDR4 are even closer.
If you’re interested in upgrading the mobo and ram, i’d suggest going with Ryzen 7 1700 and B350 with DDR4 3000. The single core performance of Intel will be better, but since games aren’t affected a ton by CPU speed, you won’t notice much difference. The 1700 is a mult-core performance monster for the money compared to 8700k. So, if streaming performance means a lot to you, go that route. This combo looks very good for ~ $200 US more than a 4790K alone, and much cheaper than building with a 7700K or 8700k (if you can snag one).
Another big plus to Ryzen for me is that AMD plans to support AM4 through 2020, meaning you should have a future upgrade path for your CPU.