Is this entirely focused on gaming? If you're willing to spend that much on a CPU and it's for gaming, to be honest you may be better served looking at Intel. If this is for a mix of stuff, including gaming, productivity that needs the cores, or even "just because I want AMD and I want a very high core count CPU" (yes, that's a valid reason in my mind), then AMD will be a very good option. Typically AMD is better when all factors are considered at most price points, but like the one time Intel is worth consideration is "I am willing to spend a lot on the CPU and the focus is gaming above all else".
The decision to make is often basically "do I want to step up to the next core count tier, or have a bit more IPS (it's not really IPC anymore as Intel just has clock speed advantage) which translates to more gaming performance?". If you don't need the cores, you can either go for Intel (or save money and get a just-about-as-good AMD offering), but I feel AMD is not only better performance for the dollar but has a better chance at longevity. Single-threaded gains aren't big anymore, so barring some new revolutionary jump there, there's less risk they fall into obsolescence by lack of raw computational power per core (and IF such a jump happens, Intel would be only slightly better off relatively anyway since the single core performance difference is so small now). But, for the "I'm willing to spend this much on a CPU so what's the absolute best for gaming", if it's above a certain price point, the answer is often Intel, even if the difference is usually small.
If you're partial to going with AMD, keep in mind that Zen 3 (Ryzen 4000 series) is probably half a year-ish away (said to release this year, but in what capacity is unknown), and those will be supported on recent AM4 motherboards. What I did was I got a Ryzen 7 3700X with the intent of leaving the option there of looking at the next generation CPUs when they release, and then if I decide it's not worth it, the Ryzen 7 3700X is still a good enough minimum for me. If you go all out now, it'll be harder to justify that maybe (or maybe not since the Ryzen 9 should still sell well after). It's really up to you, but that is the approach I took. I also went with AMD because it was ~$150+ cheaper on the CPU alone (comparing Ryzen 7 3700X to Core i7 10700K) and the small difference in gaming didn't matter that much to me. If you're willing to spend that much on a CPU though for gaming, Intel might make sense to you.
As for cooling, I agree with the above. My one point of needed assistance when doing my upgrade was with the CPU cooling, and there were three, four, five or so particular ones that kept getting recommended, and it more or less came down to those. If you do end up going with a Ryzen 9 (12 or 16 core chip) or any Intel (they run warmer I think?), I wouldn't skimp here and would go with one of the better air coolers.
"The heart has it's reasons that reason knows nothing of."