The Noctua NH-U12S is a classic and there is nothing wrong with it, but if you are wanting a more budget option the Thermalright Peerless Assassin can get you better performance for less than half the cost $34. for performance comparison I referred to the gamersnexus cpu cooler chart
For basic gaming you can probably just use your current drives but you will want to upgrade if you plan on playing newer games since they can be 50GB+ now. Modern games really want an SSD and I would strongly recommend an NVMe drive. For reference the 850 evo has a sequential read speed of 500MB/s and the SK Hynix Platinum P41 NVMe drive I recently upgraded to is 7000MB/s or 14x faster read speed. This can make a difference in loading speeds and also with pop-in during gaming. The SK Hynix Platinum P41 is $83 for the 500GB model on amazon. Most of the newer NVMe drives will have similar read performance, but some drives have issues with cache saturation when reading larger files that you should research before making a decision.
The RX 6700 XT is a strong choice for the $330 level. The RX 6750 XT is a bit more performance for a bit more money but both are great value. For comparison the performance is somewhere between a 3070 and a 3070Ti according to Tom’s Hardware
I’m not as familiar with AMD + Linux so I will let those more familiar with that combination weigh in.
I agree you don’t need a high end NVMe and most modern NVMe drives will be sufficient. I was more so trying to point out that drives have gotten a lot faster than sata ssd speeds. I don’t have strong recommendations for a specific drive but from a quick look at budget drives the Team Group MP44L 500GB is about 5000MB/s and about half the price at $44. It is still a 10x speed increase over the 850 evo. Some upgrade to NVMe would be highly worthwhile for gaming.
I was a bit mistaken the cache issue is for sustained writes not on sustained reading as I previously mentioned. The MP44L cache is large enough that you would probably never run into an issue with it unless you are re-imaging an entire disk. For drive performance I rely on Tom’s Hardware. For cache issues I specifically refer to their sustained write tests. The main thing to avoid is drives with a small cache as shown by the severe throttling of the Solidigm P41 Plus on that sustained write chart.