You may be right. A simple trick could be blowing all the dust out of the psu with condensed air or an air blower as they dust could be interfering with the electricity from passing through. Try replacing the cables that connect to your gpu, cpu and mobo as one of them could be damaged or worn. If that doesn't work you could try buying a new psu and see if that makes any difference (make sure you can return the psu if it doesn't work) .
The pc shuts down on boot-up as (most likely) an electrical surge/overload protection to protect your components due to something not working in the psu.