I5/I7/I9 differ mostly in the number pf processing threads,
Gaming depends more on the performance of the single master thread.
Run the cpu-Z bench on your i5-11600K.
You should see a single thread rating of about 639:
https://valid.x86.fr/bench/rf2mte
That is really quite a decent number.
Have you tried overclocking?
It might get you something, but chips are binned and there will be very little headroom making the effort not much worth it.
Try this experiment:
Run YOUR games, but lower your resolution and eye candy.
This makes the graphics card loaf a bit.
If your FPS increases, it indicates that your cpu is strong enough to drive a better graphics configuration.
If your FPS stays the same, you are likely more cpu limited.
You might also share the make/model of your other parts.
A significant cpu upgrade is going to involve a motherboard upgrade.