@OP
This is a very bad PSU.
It is limiting the 12V output to prevent the +5V going too high.
Recapping does not change this behavior, it is caused by bad design.
Replacing it with a good one is the only option.
@Errius
Yours suffers from the same problem.
In addition, the ripple will probably be excessive.
If you want your HDD not dying soon, then replacement is the right option.
@waterbeesje
Excessive undervolting of the 12V rail can be very harmful, as it puts a lot of strain on DC-DC converters which these are not designed for. They have to run way faster than designed for to pump the energy.
I won't be wondering at all if the PSUs being described in this thread are FSP ones, as the behaviour described is typical for FSP PSUs and many other nonames.
I know many people here who did not do actual test bench examinations are FSP fanboys, so I know that there will probably some angry reactions on what I wrote.
My personal advice is to use Seasonic, as I never have encountered any PSU of these which fails the ATX crossloading specs like the PSUs that are being discussed here. By using a Seasonic, you just cannot do it wrong.
There are other PSUs that are good like Enermax and Bestec, too, but these are either rare/hard-to-obtain or of brands that have good and bad PSUs, so one needs to do a test bench examination before deciding whether to use or discard that particular PSU.