There is a decent ammount of steam games as well that can run on intergraded graphics cards, Mostly the 2D sprite pixel games new and old, or low level 3D games, somewhere between playstation and n64 based graphics that can work as well.
Undertale is one of those low end games, is a rpg game. Grandia II is also one of those possible games, as well as one called Pier solar and the great architects, or guacemelee, lethal league, speed runners, dustforce dx,
Hunniepop maybe even some date sim or graphic novel games, or some fighting games like king of fighters 98, Vanguard princess and Yatagaratsu attack on cataclysm and maybe skull girls.
If you have steam is still possible to play games, there even a decent port of sonic CD that uses 16:9 Wide screen and crips graphics at 60 fps for Cheap. Just be sure to check the bottom of each steam game page for system requirements. The ones that should work is probably small file ones probably Under 1.5GB and require no more than 1 GB of ram.
But always check system requirements at the page before getting or try demos first. Also not all games require Xbox controller to work or what is known as Xinput. Some use common HID input.