I checked it, made everything on low and still runs with 60 fps. My CPU is at 80 °C from just leaving the game on and nothing else opened. So.. That might be the first thing to change.
Your test tells me that your cpu is holding you back for gaming, let alone also streaming.
Your i5-7400 has 4 threads and a passmark rating of 7237 when all threads are fully utilized. The single thread ration is 1945. Your cpu runs at a clock of 3.0 with a turbo of up to 3.5.
Turbo speeds happen on only one thread and only when thermal conditions permit.
80c. is ok from a safety point of view, but it may be inhibiting your turbo.
The stock intel cooler is only ok, not the best.
The conclusion is that your cpu holding you back.
To keep your current motherboard, the strongest cpu upgrade would be a 8 thread i7-7700K.
It has a passmark rating of 11940 and a single thread rating of 2581. The stock clock is 4.2 with a turbo of 4.5.
A i7-7700K seems to sell on ebay for $270 or so. (Your i5-7400 is around $110)
A i7 will likely need a better than stock cooler.
Past that, you are looking at a current gen intel or ryzen processor which will also entail a new motherboard and likely a cooler also.
On the Intel side, a i5-9600K costs about $220 and is perhaps 10% stronger than a 7700K.
You will need a z390 based motherboard which goes for perhaps $115 and a decent cooler in the $40-$80 range.
A i7-9700K has a rating of 17216/2815 and costs some $400.
Either can likely run at 5.0 on all cores with a good cooler.
On the ryzen side you get many cheap threads which is very good for streaming and multiplayer games.
Ryzen 3600X is $240. A X570 based motherboard starts at $150.
You would probably want an aftermarket cooler also.
3600X has 16 threads and a passmark rating of 20520/2916.
Ryzen comes pretty much overclocked so do not expect the added headroom you can get with the intel K suffix processors.
For pure gaming, the better single thread performance comes from Intel.
But, for multitasking as well as gaming, I think the ryzen option would be better.