My interpretation of "budget" seems to be obsolete. My budget PC:
CPU: $230 Ryzen 8600G
GPU: none
Mobo: $120 AM5
RAM: $130 32GB DDR5
SSD: $80 2TB PCI-e 4.0
PSU: $50 5-year warranty 550W
Case: $40 glass panel, w/ fan filter
TOTAL: $650
That'll run almost all games at 1080p. Then add a graphics card for $300 in a year or two, whenever your backlog of games from 2020 runs out.
If you're not compromising anywhere, it's not a "budget".
EDIT: That CPU cooler is likely adequate with decent case cooling. If you want to spend $35, this is pretty great for the price:
https://www.amazon.com/gp/product/B09GTN6ZRR/