There is no such thing as "bottlenecking"
If, by that, you mean that upgrading a cpu or graphics card can
somehow lower your performance or FPS.
A better term might be limiting factor.
That is where adding more cpu or gpu becomes increasingly
less effective.
Some games are cpu limited, others are graphics limited.
Try your games with the new processor and see how you do.
What was your old processor?
To get some idea of what the 5600x might do,
Try this simple test:
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 was already strong enough to drive a better graphics configuration.
If your FPS stays the same, you were likely more cpu limited and that your cpu upgrade will do you some good.
Most games are graphics limited. Particularly at higher resolutions.
But, most games rely on the single thread performance of the master thread.
The 19% ipc boost of the 5000 series ryzens will do an excellent job on that aspect.