That's a very elaborate setup and hardcore if you want to have a two PC setup for streaming and gaming. The computer doing the actual streaming will need a good amount of CPU power, while the gaming PC is ok with a Ryzen 5. In other words, my advice to you is to use your existing PC into the gaming PC, while you build another as the stream PC. Streaming uses more CPU than GPU.
One point of clarification is needed in your statement. What do you mean by "play games with decent graphics to make everything look better for videos"? The RX 580 should be decent at 1080p or is it not enough to your standard? Or are you pushing it to 1440p?
Or are you saying when you stream, the video quality of the stream is lower than your expectation? If the stream quality is low, then that's your internet bandwidth and CPU power lacking.