I quickly threw together a full parts list. The RX 580 is not actually $250, one can easily be found for $50-$100 on eBay, bringing the total price down to $475. It's certainly not ideal performance, but $500 doesn't go as far as it used in this RAM and SSD market. You might knock the RAM down to 8gb or the SSD down to 512 if you still need room in the budget for a monitor and peripherals

Nvidia 1080ti and Ryzen 5 5600x is a pretty solid pairing that will run you a little over half your budget. I had the very same setup until about 2 years ago. You ought to be able to run anything pre~2021 on max settings without any issue, and pretty much anything afterwards with medium-high graphics (YMMV).

The GPU will be your bottle neck here but is about the most cost effective option. Consider upgrading it if possible