From reading this thread, and a few others, I learned about taping-over pins B5 and B6 (SMBus).
Now my $15 HPE Ethernet 10Gb 2-port 530SFP+ works great in my 12 year old Gigabyte consumer motherboard.
Y'all are the people that make the internet great!
From further reading, I learned that these pins are used for power related management communication. (Like hot-swap features, etc.)
That helps to explain why, the instant I plugged in the card, my computer turned on.
In a panic, I killed the switch on the PSU, re-seated the card, and tried booting. Many attempts, and made no progress. It wouldn't even try to POST. Each time, I had to pull the card to get into the BIOS and fiddle with things. All that BIOS fiddling did no good.
So I YOLO'd the tape trick. Success!