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.
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.
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.