It is possible it is two dead batteries -- who knows how and where the amazon effort (which is most likely either an ancient replacement, a reclaim or a one hung low Chinese effort) was stored.
If the charge light stays on and the thing powers for a few seconds that does tend to eliminate the fuses and em8/em10 issue. Power switches on SPs are usually pretty good -- DS lites might be fun, and volume control on anything also annoying but power switches are not really an issue outside of specific damage (dunk it in salt water say).
Not working with the charger is possible if the battery is dead enough that it is gobbling too much current that there is not enough left over to power the thing. You can try powering it on with charger connected and no cartridge in there (some carts, especially some flash carts, can tip it over the edge into where there is a net drain on batteries rather than even a trickle of a charge going in and even without that they do take a fair percentage of the total). I don't know what we have right now to fake a battery (which does need to be there)
The charger could be at fault, though it is rare for this sort of thing. Bonus for you then is if it is bad at the adapter side you presumably have a nice version of the port connector and the voltage it takes in is... USB so grab a sacrificial USB wire from the pile and wire it up (if it is a cheap Chinese USB you will be advised to make sure you have the right wires -- the normal colour code is not respected at all).
