I think the problem might be with the "faulty 3070 GPU", i know that you have a 1060, but can you test with samo other stronger GPU, maybe from a friend or somebody?
And also could you test your GPU in some other PC?
I had an 2600K paired with 3080 until few months ago, and i played almost all of the mentioned games with lower or medium settings and did not have stutters like that and that CPU is 10 years old.
For the mentioned games, the Ryzen 5600 and 3070 should not be a problem, especially with lower setting. I play LOL on my laptop with integrated GPU and 5625U CPU and it doesn't stutter, the settings are all on low and the FPS is around 70-80 but it doesnt stutter.
In my new PC now i have Ryzen 7700 and 3080 and everything flies i those type of games, the FPS is much higher but as i mentioned, it was stutter free on my old setup as it is on my new setup, just that i now have double the FPS.
You say that this did not happen with the Intel CPU, so its logical that the GPU is not a problem, but you mentioned that your motherboard broke, could it be that the GPU also "took a hit", what kind of failure did broke the motherboard, was is some sorte of electrical/power on-off issue that borked the motherboard?