If the 'shorting' works for you then it's the pc-case power button faulty, probably. just for a check, switch the restart cable with the power off/on cable (if that works even). then if you can power up with the 'switched' restart-to-power-up button, it may be the cables/wires fault.
take them out, replace them. if you're okay with soldering a new one, you're good to go. if not, try another pc-case or, a problem less, find someone who can solder the new power button wires. you can use an old router to LAN cable to get the wires, just remember which color is negative and which is positive. hope it yelps.