Make and model of the PSU and it's age? You might want to relocate the system to another wall outlet and see if that helps power the system up.
As a side note, the paper clip test isn't a false test. The paper clip test asks you to induce a load on the PSU, using either a case fan or a dead/non important HDD, this doesn't indicate how much power the PSU can effectively output to the entire system even though it shows that it can power up.
On another note, please list the specs to your entire build so we can gauge if your PSU is lackluster or inadequate for the task.
Might want to also shoot us a picture of the black piece you found in the case. It could be a capacitor that got knocked off the board, in which case, either the board is...