While the system is a prebuilt, the specs of it are pretty admirable:
Ryzen 5 3600
32GB DDR4 (looks like you added another 16GB)
NVIDIA RTX 3060 Ti
512GB PCIe NVMe SSD (plus the other storage you added)
Only issue I really see is that it appears your memory isn't running XMP, assuming when you upgraded and added the additional 16GB that it also was 3200MHz. Really all you could do is upgrade to a Zen 3 5000-series processor, assuming the motherboard included in the prebuilt has a BIOS update to accept it. Looks like there was a BIOS update released on April 2022. Would have to check the ReadMe to see if that's what its purpose was for (adding additional support for newer CPUs). It seems the maximum PSU these pre-builts offer is 650w. Might have to specify what your unit came with as it appears there were a few options (ranging from 450w to 650w). Upgrading to a better PSU unit makes sense if you also plan on upgrading your GPU. Although, there really isn't much more you can do upgrade wise for your GPU other than an RTX 3090 (an RTX 3060 Ti is quite fast). With next-Gen GPUs (NVIDIA Lovelace 4000-series) right around the corner you are better off waiting until then to upgrade your video card.
As far as moving the pre-built into a new chassis/case, you have to be careful with that because many of these pre-builts have proprietary connectors, motherboards, and other nonsense that makes it difficult to do. This is some research and investigating you will have to do yourself. If you aren't experiencing thermal throttling, not sure if upgrading the CPU cooler is necessary either; Ryzen tends to run pretty well with relatively basic CPU coolers to be honest. Not saying it doesn't make sense if you upgrade to a better CPU, but generally a good air cooler will get the job done. Repasting isn't a bad idea, might be good for a few degrees here and there; with some high quality paste (I recommend GELID GC-Extreme or Prolimatech PK-3; different people will have their own recommendations).
Besides that, I think I covered the majority of your questions/concerns.