I wouldn't put a $250 CPU cooler on a build of this level (or on anything but the highest-end hardware, for that matter). Even if you planned to overclock, the real-world performance benefits would be minimal compared to running the processor at stock. And you could get more performance while spending less just by going with a higher-end CPU. Even at stock clocks on an inexpensive tower cooler, the 12600K mentioned above should be faster than an overclocked 11600K. Or even the locked 12400 or a 5600X might be a reasonable alternative to an 11600K, though the 12600K's additional low-power cores will also provide more multithreaded performance for heavily multithreaded software like video encoders, or potentially future games that might benefit from them down the line. I wouldn't spend over $100 to cool any of these CPUs though, and even most $50 coolers should work fine.
And while the 12th-gen CPUs do offer motherboards that support DDR5, any semi-reasonably priced kits of DDR5 will be outperformed by DDR4 in games, making them a poor value, so I would stick with the DDR4 models. DDR4-3600 or 3200 will likely offer the best value to performance ratio, and while you might be able to get away with 16GB today, you may want more for multitasking, and it probably won't be too long before some games need more for optimal performance, so 32GB is probably best for the long-term. The extra RAM should be a better use of funds than an expensive cooler, at the very least.
And of course, a better graphics card will contribute more to performance in most games than anything else, so shifting more of the budget toward that might be a good idea as well.