No, it's really not.
Your stuttering is probably a lot more likely to be a result of only having 8GB of RAM than it is from a lack of CPU resources. Your 4770 has better single core performance than the Ryzen 2600 and your 8 threads are more than capable of handling any game without stuttering. That's not to say that additional cores aren't helpful, I just don't believe that your problem stems from a lack of cores.
If you're going to upgrade the platform, which would mean an entirely new motherboard, CPU and memory, no matter what you upgrade to unless you put a 4790k in there which would basically be pointless, you might as well upgrade to something that's fairly current. Not something like the Ryzen 2600 that's already 2.5 years old. Intel Coffee lake refresh and Canyon lake parts are good. Ryzen 3000 series is good and the 5000 series parts are getting ready to release not long from now. If you're going to buy something, those are what you'd want to buy into.