Short answer: A $300 budget cannot build a new PC that runs modern AAA titles (Cyberpunk 2077, Elden Ring, Warzone) at 30-40 FPS on Medium settings. Current GPU prices alone (e.g., RTX 3050 / RX 6600 ~$200-250 used) exceed the budget when adding CPU, RAM, SSD, PSU, case, and OS.
Save to $500-600 for a true entry-level 1080p AAA build (Ryzen 5 5500/5600 + RX 6600/7600). Until then, a used console or cloud gaming offers a vastly superior experience.