So I bought a new PSU and interesting things happened. First I got the exact same symptoms as with the old one. Then I took some components out to reduce the possible errors. When I unplugged the power cable for the SATA ports that lead to the 2 disks it suddenly worked again. Then I put it back in and it still worked. I have not yet checked the old PSU but I assume that it wasn't its fault and it's still working properly.
My latest deduction is that there are bad contacts on the SATA data ports which caused a short circuit, which then prompted the PSU to shut off. That perfectly lines up with the symptoms. When I turned the PC on it worked for a split second and then immediately shut down again as a safe mechanism when it discovered the short circuit.
The problem with the initial troubleshooting was that it instantly worked with the spare test PSU which let me to believe that it's definitely the PSU. But it probably only worked because I did not connect the disks.
But now I'm too lazy to investigate it again but the conclusion seems sound. I will investigate further when or if it happens again.
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