The GIGABYTE GA-970A-DS3 motherboard is broken because
Gigabyte did not qualify the hardware that they used.
The Etron Technology, Inc. EJ168 USB 3.0 Host Controller (rev 01)
does not properly handle USB hubs.
By adding a PCI-E controller card:
Renesas Technology Corp. uPD720201 USB 3.0 Host Controller (rev 03)
devices plugged into the USB 3 hub are seen and enumerated correctly.
The down side, of course, is that the BIOS will not see these devices.
Consequently, to change BIOS settings, I will need to manually replug
my keyboard into another main board USB port.
I will live with this until either the main board dies or hardware advances
make it otherwise advisory. The next board will not be a Gigabyte.