Yes, you can use a USB game controller without root and without needing any prior setup. Just plug it in and Android will recognise it. By default, the D-pad on the controller will work like a D-pad or trackball on the device itself (if you remember when Android phones used to have little trackballs built in), and it will allow you to move focus between controls in most apps.
Stock apps are completely accessible using the controller without the touch screen, but not all third-party apps are. Many games (especially emulators for games consoles) support using a Bluetooth or USB game controller, but not all games do. The idea of apps like the one you mention is to make button presses on the game controller trigger touch screen events, so if you play a game that has on-screen controls but doesn't support controllers, the controller will trigger the game's on-screen controls. Generally you'll get a better playing experience, as well as easy setup, by choosing a game that supports game controllers.
If you just plug the controller in, it'll work in any supported apps and games right away. Games with controller support usually advertise it in the description on the store, but if you're not sure you can ask the developer through their support email address.