One thought, as a non-gamer (so take it for what it's worth... next to 0)
- if considering spending money on 2nd PC, might you be better off spending instead on main computer? and/or simply get gaming PC setup up adequately?
Gaming rig still has to run game. Running real-time video encoding for streaming can be demanding, and may be too much, *if* gaming rig at or near overload to begin with. Otherwise, with GPU encode offload and a CPU-considerate setup of OBS Studio, OBS Studio wouldn't add that much to gaming rig load. The issue is when people add CPU intensive load (chroma-keying) or other CPU intensive filter/effects (or really poorly written plugins... looking at you, streamelements)
The gaming rig still needs to get video out to capture (OBS Studio) rig when using 2nd PC. Depending on how exactly you do that may r may not create its own extra load on the gaming rig
For upper-end PCs, with nicely optimized Operating Systems (OS), looking to absolutely max out gaming performance with gaming PC still near 100% load, then a separate streaming rig may makes sense. and a couple of other scenarios... but otherwise, it has always stuck me as way more complicated (and therefore prone to problems, me being of the K.I.S.S. school of thought) than need be.
I'd do some OS optimizations, then look at OBS Studio setup. As I said, not a gamer, so not my area, and AMD GPU's not known for playing well in the H.264 streaming world (AMD's intentional under-effort in associated code). But with right setup, I'd think AMD GPU should be ok (but I'll let others speak to this way more authoritatively than others). Oh, and try testing without OBS' Studio Mode turned on (can be 2X rendering workload)
As for streaming at 60fps, with decent video quality to make it worth it, hopefully you are using wired Ethernet (not WiFi, unless in a rural area (lack of other nearby WiFi routers/access points) and well spec'ed, configured, and managed WiFi home environment). And solid Internet connection, being aware of ALL internet bandwidth consumption (to avoid unexpected bandwidth contention). And avoid known problem like running monitor refresh at 144MHz which is not a clean/easy multiple of desired frame rate (or worse for streaming, multiple monitors at different refresh rates).
And beware that not all games use the same video rendering approach, with means video capture settings that work for one game may not be optimal (or work at all) for another. again, not my area, but I've seen many such a thread over the years. Most folks look for a streaming setup that will work for most everything, or at least that which they'd want to stream, and call it good enough (as the content is usually more important than absolute best stream 'look'). What typically happens is either a streaming has some perfectionistic tendencies (like me) and will educate themselves on settings tweaks to improvements things, and will over time improve their setup; or a person will be more focused on content, find a streaming setup that is 'good enough' and focus their efforts someplace else other than stream setting optimization. The caution is to simply be aware that for those in the latter group, Operating System, Game, and other changes will happen over time regardless, so while your setup may work fine now, next year it might not... and that is normal (even if frustrating ... nature of the technology involved).
