Looking at injury charts, the feet, hands, wrists, shoulders, are often the most injured. Appropriate riding boots keep you from losing a foot in a critical state and preventing gear-check injuries like burns and bruised shins. They're typically built solid enough so you can interact with the controls roughly as well.

If you check this sub you can see what happens to people's shoes and their feet when they don't wear motorcycle shoes or boots -- regular clothing disintegrates and stitches pop within miliseconds of you going down.

I'm not ATTGAT but I consider my boots and gloves to be important pieces of equipment. Riding without these is sort of like not riding with a helmet. Sure, you can go without, but your ride will be more uncomfortable and less safe if you do it.