OP, I'm not sure what all lengths you're willing to go to - however, when searching your motherboard + the 970 I've found multiple people online with that specific combination and no problem, so I don't suspect it being a motherboard issue.
You mention no BIOS updates since 2013 - however this page shows one from 2014:
http://www.gigabyte.com/products/pro...?pid=4288#bios
May be irrelevant.
I'd start by removing all non essential components. Hard drives, all PCIe cards, etc. Remove all but 1 stick of RAM. Reset the BIOS using the BIOS reset jumper.
http://www.overclock.net/t/1398373/g...#post_20135949
Now throw in the 970 or 1070 and give it a try. See if you can get into the BIOS. If you can, great! Now slowly add each part back into the mix. If you still can't, it may be worth it to contact Gigabyte or NVidia support.