Probably BIOS issue.

Well, I updated my TUSL2-C BIOS to version 1014.001, the problem persisted. Then I flashed the BIOS of the Radeon to this one, still the same problem. Is there even an solution to this?
By the way: When I start the computer, the speaker beeps "Video card not found or video card memory bad" according to the manual.

With the card being up to 16 years old. The capacitors might be dying or drying out.

What PSU are you using and whats the 5v/12v output ratings?

This is the rating of the PSU:

I detached everything consuming power from the PSU and the mainboard except for the CPU and the 9700, but it still won't coldboot. I'll do more research to find out if the PSU is to blame. The caps on the card are not bulging, is there any way to check them like with a multimeter?

Update:
According to this german article the 9700 Pro draws ~150 Watt in the BIOS screen and probably even more on startup. So regardless to which rail the 9700 is connected, my PSU does not provide enough power I guess (or can it draw from both rails at the same time?). I'll connect one of my modern PSUs and see what happens.

That's full system consumption. But your PSU is weak in 12v rail department.

That PSU is a stretch for anything above a basic P3. Also, it's possibly not good for it's rated amps.

I replaced the PSU with a 450W bequiet! one, which did not solve the problem. In the meantime I managed to snag another 9700 Pro of eBay, this one works flawless. I'd like to replace the caps on the non-working one, but I'll have to get some proper tools first. Anyways, thanks for your suggestions.