Have you tried programming directly using the JTAG interface on jumper J17? If you have that option it would be my next port of call to rule out the USB - JTAG interface.
Looks like xsct can see the board, but with errors on the JTAG boundary scan. Take a look at some of these threads....
Looks like you need to verify the JTAG TMS, TCK, TDI & TDO lines are all in the the correct states and being pulled up properly with the proper voltages, etc. The issue could also be the NC7SZ66P5X which is a bus switch that the TDI & TDO lines go do, presumably to allow interface either via the FMC connector or via normal JTAG header.
However, just incase it is something this simple, you have double checked all the jumpers on the board are in the correct position, yes? Not set for say direct JTAG interface programming when you are trying to program over the USB or anything?
If the jumpers are all correct then it's time to sit down with the board, schematic and an oscilloscope / test gear to find the fault (which could still be under the FPGA on the BGA connections).
Schematic can be found here... https://reference.digilentinc.com/_media/reference/programmable-logic/nexys-video/nexys_video_sch.pdf