@Robchil I was looking at other SFF cases online. Most of them have riser cables. A few of them had notes that said "If your card is a PCIE 4 card, make sure to change it to PCIE 3 in the bios". I tried doing that on my computer, and the fans work again! So! The solution to this entire thing was for me to go into my BIOS and change the PCIE slot from 4 to 3.
That riser you linked is PCIE 4, I'm guessing the riser that came with my case many years ago is PCIE 3, so getting an upgraded riser is the fix. I wonder if it is worth the $60 to upgrade from PCIE 3 to PCIE 4 for a GPU...