The best cpu above for strictly gaming is the 9700k because of its high clock speeds, 8 cores, and 8 threads. It would see a bump in performance over your 7700k because of the 8 cores rather than 4 ht, but for some games such as pub g that aren't optimized for many threads, the improvement margins are small. Your 7700k and 9700k both are based on similar architectures, both have 8 threads, and when the 7700 is overclocked the clocks are similar. The 9700k is ok for streaming and editing but it has the least threads of any of the listed CPUs, which makes it last in these 2 categories.
The AMD Ryzen 7 2700x is the best all-rounder winning in 2 of the 3 categories, and coming close in the 3rd. The ryzen 7 2700x is the best streaming cpu of the 3 above, because the 2700x is the only cpu that offers 16 threads allowing the workload of streaming and gaming to be divided to 8 threads each. See this video on the 1800x for an explanation of why AMD's ryzen 7 CPUs are best for streaming:https://www.youtube.com/watch?v=jludqTnPpnU The 2700x is good for gaming while not streaming, but Intel wins there by a fair margin due to higher IPC and clocks and since current games don't utilize the cores to the fullest. The 2700x is by far the best for editing due to the 16 threads dividing the work up. Best 2 of 3, not bad.
The I7 8700k offers 6 cores and 12 threads, as well as high clocks. The 8700k is middle of the pack in all categories, making it well rounded. Its 12 threads make it perform middle of the road for streaming and editing, giving a boost over the 9700k but still behind the 2700x. The 8700k is very good for gaming, but its clocks are slightly behind the 9700k. For games optimized for more than 8 threads, the 8700k is better than the 9700k and about = with the 2700x. It is better than the 2700x and behind the 9700k for games that dont use more than 8 threads.
In conclusion, the I7 9700k is the best for gaming but last for the other 2 categories so id put it last. The 8700k is middle of the pack for all categories and a very well rounded cpu so it comes 2nd or 1st depending on how important streaming and editing are. The R7 2700x is best for streaming while gaming and editing, but last for gaming without streaming. The 2700x offers a good platform price, included RGB beefy cooler, and a better upgrade path than any of the above CPUs. Because of this, I crown it the winner.
All of the listed CPUs will work and be capable of your needs, so feel free to use my info above and draw your own conclusions. Also, the 2070 is a good GPU, capable of fulfilling your needs, so i would recommend it.