I have had a 4k panel since 2012, in my experience gaming performance is going to depend heavily on the gpu, the cpu is only required to run the game and deliver at least 60fps for your 60hz panel.
In 2012 I used a 3770k and now I'm using a 8700k, which is still today a very fast gaming cpu and both run 5700 xt's. In the vast majority of of games I wouldn't notice between the two, the gpu is the bottleneck in both systems as even the old 3770k is good enough to push the card at 4k. If you had a high hz monitor 120+ at around 1080p then this would be a different story, you would need a strong cpu to deliver the data the gpu needed to achieve higher frame rates.
Any decent new cpu from intel or amd will do the job, go for the one that offers you the best bang for your buck, buying a more expensive slightly faster cpu will likely only make a slight difference in cpu demanding titles, hopefully with a decent new cpu this shouldn't be a problem for a few years.