From personal experience, just an 8600k build is dramatically better than a 3770k. In the same gen, HT gives you 0-15% average improvement over non-HT with the same core count, being generous let's apply 20% over a pure quad core value for the two extra threads.
So let's say a 3770k Ivy base core has a performance value of 1.0 as a baseline.
4.0 score + 20% = 4.8
Coffee Lake IPC is on average 15% or so faster than Ivy, at the same clock speed. So that gives us a per core score of 1.15.
6 cores at 1.15 = 6.9
That's measuring with typical stock ram settings, regular air cooling, an conservative clocks eg 4.5Ghz for each.
That does not include the serious QOL improvement from a modern platform with native M.2 PCIe SSD support, support for hugely faster DDR4 packages (I'm running Patriot 3733 @ 4Ghz with my 8086k), improvement to PCIe, USB, and SATA performance, common USB C/3.1 support, yadda etc.
I only recently retired an outstanding 4790k build, and although the 4790k is absolutely still incredibly good, I'm already seeing platform improvement thanks to a few gens of these kinds of features. I also quite like my Ryzen builds, but don't use any of them for gaming. I won't rule out the potential for a drop in next gen Zen chip being my next gaming platform should it become better than a 5.2 8086.
For a gamer and emulator, particularly with newer emus I'd really recommend a high speed Intel. Even an i3 K series at 4.6+ is stellar there, though I'd push for an i5 or better for longevity. But you're talking about multiple great choices. 2600 or 8400 or better, and honestly you should be in great shape. Hell, if you save some bucks with a Ryzen build, then you could potentially be able to toss an m.2 nVME SSD into the budget