Fuses don't work like that, once they blow the item they are protecting will stop working. There are fuses that can trip and reset but I doubt the Wii has any of those.
If the Wii is coming to an on power state and failing once loaded then I would be looking towards overheating issues. The Wii doesn't produce all that much heat, after all the total power usage is only 18 or so watts, so I'd imagine that it's not a dodgy fan as that would take several minutes to cause an issue.
It could be a dodgy heatsink on CPU/GPU. It uses an passive aluminium heatsink sitting atop both chips, if this was loose and not making a contact on either or both chips then they would shut down very quicky. I'm not sure about the retention mechanism for the heatsink, it may only be epoxy glued.