Faulty RAM sticks have become so rare that I don't test salvaged ones any more.
As for running a memory test, a failing power supply will also cause the test to fail. Since you had to take two sticks out, a power issue sounds plausible. I'd run the test on another computer if possible, or have another power supply at hand. Start with all four on to see if the test fails. If not, run the test through several times. If it fails, this would be the time to do the same with another PSU. If you don't have one, run one round with only two sticks. If there's no issues, reseat the sticks to the empty locations and run the test again. Then, if it's still successful, do the same for the other pair. In case of success, run the test with three sticks, again changing their places to find out if it's a certain slot.
As you can see, there's several "stone plates" on a memory stick. If one of them fails, your system may boot and run just fine until that certain "stone" gets used, in layman's terms.
9 years for a PSU is quite a long time, especially if you've changed your video card in between. There's been some design changes in the 12V line during the last decade or so. How about getting a new rig in smaller portions, starting with the PSU?