With a 144Hz screen that can display higher frame rates, the Intel CPUs can potentially have a slight advantage in current games, perhaps by as much as 10% or so when performance isn't being limited by your graphics card, or a bit more if overclocking. However, in most of the bigger recent titles at high settings, even at 1080p, the GTX 970 is likely to limit performance more than anything once you start looking at those higher frame rate ranges. Most recent CPU reviews test with something like a GTX 1080 or 1080 Ti, while with a less powerful card, the differences in frame rates will tend to be quite minimal between any of these CPUs in current games, much like when testing with a higher end card at higher resolutions. The GTX 970 is still a pretty good card, but it's unlikely to push the most demanding games up into the 100+fps range where the CPU becomes more of a limiting factor.
So, the question becomes more about future performance. I suspect you'll probably want to upgrade your graphics card sometime in the years to come, so it's certainly possible that the slightly higher per-core performance of the Intel chips will give the graphics card a bit more room to stretch its legs once you do. And of course, games will become more demanding, and perhaps in a few years, that bit of extra performance will help, even for just maintaining a steady 60fps in some games. However, another thing to consider is that games are becoming more multithreaded, and while the 6 cores of the 8600K should be great for running games now, it's possible that games will become more heavily multithreaded over the next few years, allowing them to benefit from having access to more physical cores and/or threads. The 8700K has 6 cores with SMT (Hyperthreading) to more efficiently run up to 12 threads, while the 2700X has 8 cores with 16 threads. Right now, gaming alone doesn't really utilize these extra cores and threads, but it could eventually, especially if the new consoles coming out in a couple years or so have substantially better CPUs than they have now, causing game developers to increase demands on PC CPUs as well. Right now, the PS4 and Xbox One both use older-generation AMD CPUs, that are actually slower than what you have now.
Ignoring cost, I suspect the i7-8700K might provide the most longevity out of these CPUs. The Ryzen 2700X does have two more cores, though clock rates are a bit lower, so there's a tradeoff there, and I think the 8700K's higher-clocked cores will provide a bit more balance between core counts and per-core performance as far as gaming is concerned, even if the 2700X can potentially offer a bit more performance in some heavily multithreaded (currently non-gaming) tasks. The i5-8600K will likely have a good run as well, though I can't help but think that eventually just have 6 cores without SMT might result in less stable performance in some future games. There's a lot of speculation here, of course, since a lot of this depends on what game developers will be targeting a few years down the line.
Taking cost into consideration, at least going by US online prices, the AMD processors perhaps become a bit more attractive, with the 8-core, 16-thread 2700X currently on sale at Amazon for just $20 more than the lowest price you can find a 6-core 8600K for. Plus the Ryzen processor includes a relatively decent stock cooler if you don't already have a suitable third-party option, while Intel's unlocked "K" processors don't come with one at all. And due to Intel's current production issues, the 6-core, 12-thread i7-8700K is currently around $80 more than a 2700X, and again, doesn't include a cooler.
If you don't intend to overclock (also requiring a more-expensive Z-series motherboard for the Intel chips), another processor worth considering would be the locked i7-8700, currently available for just around $10 more than the 2700X. It's clocked very close to an 8700K at stock clocks, and is typically only around 2% slower when other components are not limiting performance. Being a locked CPU, it can't be overclocked, but can generally be found for a significantly lower price than an 8700K, and will still tend to perform better than the 2700X in most of todays games, again, provided the graphics card isn't limiting it. There's also the lower-priced 6-core, 12 Thread Ryzen CPUs like the 2600X (or the 2600 with an overclock), which can provide close performance to the 2700X in today's games that aren't really utilizing the extra cores.
While budget might not really be an issue, the money saved could perhaps be put toward upgrading your graphics card sooner than you might have otherwise.
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