Here is an update, in case others have problems with Gigabyte motherboards.
My problem seemed solved, after various tinkering, but eventually kept coming back.
I discussed the problem with a technician at the store, and he indicated that others had experienced similar or related USB/boot problems.
He said that it has been reported that these motherboards will often recognize a USB device once, but then fail to recognize it afterwards.
I was trying to boot with 2 powered USB 2.0 hubs connected (I also plugged a wireless wi-fi dongle into one of the USB 3.0 ports and left the other one empty). One hub hosted 2 printers and a scanner, on the other side of the room (hence the need for power) and the other came up onto my work desk, on a short lead, to accommodate my keyboard(s) and mice.
My problem was solved by replacing one of the powered hubs with an un-powered one. Apparently, the second powered hub, with my I/O devices attached at boot, was confusing the motherboard. When I went with a simple, un-powered, 4-port Toshiba hub, it seemed satisfied.
For reference, I had been using a pair of cheap Chinese-made powered USB hubs in this way for over a year, with my Asus motherboard, with no problem. My first attempt was to replace the hubs, and I used a square 4-port hub by Inland for the printers, and the longer black 7-port hub shown below for the keyboards, etc. This one looks identical to the previous Chinese ones but was branded Inland and sold retail at Micro Center.
I had previously liked these hubs, they are rated at 5V, 2A, which seemed pretty good. Maybe my understanding is faulty, but I thought that each USB connection was limited to 500ma, and so even with all my ports populated, each could still draw nearly 300ma.
Could it be possible that with 1 keyboard and 1 or 2 mice connected at boot-up, the hub was actually too powerful, and was sending the mobo into some sort of panic?