I presume the system works fine with the old card? Have you tried the integrated GPU as well? If the system works with the other gfx solutions, it appears to be a bad card. There is one other test you can do if you feel it's worth it...
Breadboard testing.
Remove the board from the case and set it on an insulated surface. Connect just CPU/cooler, memory, monitor to iGPU, PSU, and KB. Momentarily short the two pins that the power button would connect to for starting. See if you get a display and can enter BIOS.
If so, then try the GTX 1060 with its 6-pin connected instead of using the iGPU. Be sure the card is fully seated and the 6-pin is firmly inserted so that the clip locks.