Most games are limited by the graphics card.
Good luck in finding one at any price, let alone <2x MSRP.
Any modern processor in the $200 price range can give you good gaming.
If you are running batch multithreaded apps, then many threads are good.
Many threads are also good if you play multiplayer games with many participants.
Otherwise, 8 threads(4 cores plus hyperthreading) is sufficient.
Extra threads do not help much; only if you are also multitasking.
If you have a budget limitation, I would look at the i5-11600K.
It has 6 cores/12 threads.
What I like is the better IPC. Most games depend on the performance of the single master thread.
ryzen 5600x would be similar.
Pay an extra $20 or so for the K which includes integrated graphics.
That lets you test while waiting for a graphics card and can keep you running if graphics should fail.
Use a Z590 based motherboard.
I am starting to think that overclocking is not needed anymore for gaming.
Let the cpu and chipset use turbo to ramp up performance when needed.
The limit may be the cooling capability you install.
I recently installed a I7-11700K on a asrock Z590 extreme.
Initially I tested with an old noctua NH-12 cooler.
I was surprised that running a simple cpu-Z stress test that temperatures peaked around 65c.
All settings were at default excepting setting ram at XMP speeds.
These chips do not run hot.
There is more for me to learn.
There is an option to disable some limits, and that may be what drives up voltages and heat.