Power cables, especially barrel connectors, can fail somewhat short or send things up the wrong way -- might want to poke a meter into the end to see what it says, pressure the strain relief and see what it says then (use pins to gain access if you have to). Do also check the bottom of the charger hole for any tin foil. For the meantime all PSPs should support USB charging, though you might have to go into the settings to sort it.
As above sometimes it can be a loose screw, sometimes it can be something shorting out against a case, other times it can be a short on the board that the pressing of a button or turning it upside down will cause to activate a short. I don't think this will be something coming loose and something else dumping current (see why you are told not to remove inductors in operation) but that is a further option for some things.
I have not heard of this happening on a PSP but at the same time I am not really in the weeds of PSP component level repair. Have seen it happen on any other number of boards over the years as one of the more annoying but repairable options in such things.
Depending upon where the fuse is then you might be able to use that to narrow things down (if it is before the fuse that is one thing, if it is after the fuse that is another). Otherwise bumping is OK but tends to trouble everything. Might want to instead make a minimal viable booting thing and then apply force to various points on the board to find the most likely candidate. From there visual inspection and current measurements as you chip off sections of track to find whatever bridge, via or part of a layer is causing you fun.
There is such a thing as a resettable fuse as well that might help in diagnostics/narrowing down location rather than soldering new ones every 5 minutes but they are expensive and will likely be a special order.