Integrated graphics are not powerful enough to run AAA games at maximum settings but are sufficient for low-mid end gaming. Besides on a low budget below US$1000-1250, stick with AMD.
Settle with a mid-high end AMD APU then the lowest end Intel CPU, better performance per dollar is obtained. You also do not need upgrading potential, since people that buy budget builds rarely have money to upgrade every year or two.
You can single-source the parts from Amazon.
With less than US$400 (cheaper than a console), you get a build capable of both 1080p 30FPS or 720p 60FPS gaming and also can be used for computing tasks like web browsing, office software and video streaming.
PCPartPicker part list /
Price breakdown by merchant...