You probably have a brick, as in a corrupted BootROM/SPI flash. Download the Apple Technician Guide for your 2010 Mac Pro, go to the part of the Minimum Configuration Testing and check the diagnostic LEDs, around page 28 to 33 if I remember correctly.
See what LEDs are lit and what are off, then follow the diagnostic steps that Apple suggest.
With your described symptoms, my bet is that your EFI DONE diagnostic LED is off and you have a brick.
GPU defects can make a Mac to not boot in a way that is very similar to a brick, check with another know working GPU before trying to desolder/reprogram the SPI flash or buying replacement backplanes/Mac Pro MATT card.