What kinds of games do you play?
If you play fast action games, then a good upgrade might be a stronger graphics card.
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 is strong enough to drive a better graphics configuration.
If your FPS stays the same, you are likely more cpu limited.
If you are cpu limited, a processor upgrade makes sense.
What is the make/model of your motherboard?
If you have a z390/z490 motherboard some 25% added cpu capability is available via overclocking.
If you will also change the motherboard, a 5600X makes sense.
The higher IPC is impressive. The single thread performance is what matters to most games once you have 4-6 threads.
12 threads is probably more than you can effectively use for gaming.
Multiplayer excepted.
As you said, 5000 series processors are hard to find at any price, let alone anywhere MSRP.
Unless you are in a hurry, I would wait to see what intel rocket lake brings.