So I submitted a ticket somewhere to AMD-4u and someone responded fairly quickly with most of the basic suggestions: power, seated firmly, try different connector, etc.
I hadn't tried updating the Gigabyte mobo BIOS so I downloaded that and flashed it and everything seemed fine: it still started up, the i7 graphics still worked, I can hit the internet, etc.
Then I go and do a full uninstall of Catalyst, reboot a couple times, and try running the .exe again. Seems fine, but after the reboot I have no options for the 7970, and no right-click menu item at the top for Catalyst. I go into the uninstall and Catalyst says nothing was installed inside of it - no drivers, no utilities, nothing. I tried uninstalling various ways but had no luck.
Then thanks to tomshardware.com for pointing me to the http://sites.amd.com/us/game/downloads/Pages/catalyst-uninstall-utility.aspx though in the end it looked to just delete a few files/directories which Control Panel > uninstall didn't do.
Toms also pointed me the way to the autodetect utility (http://support.amd.com/us/gpudownload/windows/Pages/auto_detect.aspx), downloaded and ran it and the first thing it says is "We were unable to find your product or OS" even if it does think the graphics hardware is "0x8086 - 0x412 PCI".
I also tried re-downloading the 13.4 catalyst, but that's the same size and shape as the previous ones I'd downloaded (which worked waaaaay back at the start). And has the exact same outcome of installing only Catalyst and Catalyst then not thinking any drivers or other utilities are installed...
This is a brand new card which I'd used for 6-8 hours on very light 3D, and on the next reboot seems to degenerate ever since with stairs and other color artifacts - which is before doing anything with the mobo BIOS.
Anyone have some ideas what's next to try after my computer doesn't recognize there's a 7970 installed in it?