Have you already tried cleaning the fan? Maybe a lot of dust is blocking the fan so the notebook can’t start and turns off itself due a higher temperature. It would be worth a try cleaning it.
But in my opinion you have everything done what you can do and it looks like a mainboard malfunction. If cleaning the fan doesn’t help I think the mainboard is defective or something else. If you are familiar with notebook disassembling you can try to repair it yourself with second hand parts. Otherwise it’s time for a new notebook I think…
I think this could be a serious hardware problem because the notebook shuts down after 5-15 sec after booting up.
I doubt it could be related to a higher temperature since the temperature could not reach a critical level in such short time (5 sec).
I think it’s something wrong with a part on motherboard but this is just an assumption.
Since this is a oldie, I don’t this its worth to replace a motherboard.
The part is very costly and maybe a second hand notebook would be a better choice…