With the Dev Boards, you could send long descriptors to PS 3 via USB. In very long descriptors, the PS3 reacts with buffer overflows, which was used in the past, to smuggle Code onto the stack from the PS3 and start code.
For the Bad Usb Exploit, in the first the exploit overwrite Stock Firmware on USB Controllers with Exploit Code. I am working in a high-performance computing Datacenter. We have computers whose USB and other Hardware controllers are accept only firmwares that are encrypted/keyed. I reckon, it will be the same on Xbox One.