Will the be quiet! Pure Rock 3 adequately cool a Ryzen 7 5800X3D?

Yes, it should be fine for your use. The 5800X3D can run fairly warm because of how the extra cache is positioned over the CPU die, but gaming loads are much lighter than a sustained rendering workload. Make sure the cooler is mounted properly and your case has decent airflow.

A Pure Rock 3 is rated for substantially more heat than the processor typically draws, so I’d use it without worrying. A stronger cooler could reduce peak temperatures and fan noise, but it isn’t necessary just to play games. An AIO would be optional rather than required.

You’re close to the cooler’s practical limit under an all-core stress test, but normal gaming should be completely manageable. Don’t expect dramatically lower temperatures from a huge cooler—the 5800X3D tends to have diminishing returns because of its chiplet and cache design.

A similar Ryzen 7 5800X3D stayed below 80°C at peak with a Hyper 212-class air cooler. Expect roughly 75–85°C in demanding games or around 120W during a heavy stress test, depending on the case and room temperature. A good thermal pad or quality paste can also make installation easier and more reliable.