In most cases it will make little difference to get a better CPU.
It totally depends on the game and settings chosen.
You can TEST this by:
a) disable VSYNC, and
b) monitor "GPU usage" (MSI Afterburner, EVGA Precision etc. sometimes hard to figure out how to show that onscreen)
If you get over 90% then the CPU is either a very SMALL bottleneck or not at all. Run Unigine VALLEY. The GPU should stay near 95% except when switching between new test areas.
Other:
I had an issue where in POWER OPTIONS the PCIe link-state was set HIGH and throttling my GTX1080. check that.
There are other reasons such as video drivers etc but we need more information.