Check the pads to see if they connected to the 5V rail. If so, it's probably just a filter cap. Just solder a 22uF 16V 0603 SMD cap.

Indeed the pads are directly connected to the +5V and GND of the floppy power connector. But isn't a 22 uF a bit high value? Especially in a 0603 size. I'm used to 100 nF in these spots.
Not an electrical engineer, just stating my gut feeling.

It's probably 2.2uF.

See the reference schematic from gigabyte. 22uF were way bigger in those days. You can see them around the memory.

Since that SMD cap shouldn't pose any problem, I gave it a go in this state and I got no image.
I even tried with a 2nd PCI card and no image on the PCI card if the Radeon is connected. No shorts on the 5V or 12V.

Post analyzer card stops at 1d 1C. Nothing immediately obvious except from a few electrolytics banged in the head. Maybe that's the problem.

You can check some of the voltages on that card, to see if the power delivery works correctly:

Re: ATI 9700 Pro Broken

Those are all the ones for the memory.

Does the GPU get warm?