I have yet to properly go through the GBA SP charging circuit.
Generally though.
The SP uses a 3.8V Lithium Ion battery
The charger in is either 5.1V or 5V if you use a USB thing.
Equally one does not put a voltage there and wait if you want your battery to last and not blow up --
http://batteryuniversity.com/learn/article/charging_lithium_ion_batteries
To that end the SP first has to drop the voltage and second has to manage that curve in the link above (there are simpler versions but not by a lot). This takes extra circuitry between the port and the battery, said circuitry almost by definition being fairly high power.
If there is a fault in the circuitry then it is not going to charge. Equally continuity might not be detected (5V to 3.8V implies a fair bit of resistance that might not play with some meters). We run into my limit of knowledge here -- I do intend to properly go through the charger side of things for the SP for
https://gbatemp.net/threads/thinkin...gba-hardware-fixes-compilation-thread.485501/ but have not gone there yet. Equally while the DS has a fairly notorious for failing fuse the SP does not have such a well known failure. Hopefully the DS stuff is similar to the GBA stuff (2003 was the SP release, 2004 was the DS with the same port and same style of battery so I am hoping).
As others mentioned the charger ports are pretty good. Damage tends to be obvious, and either in the form of corrosion or obvious mechanical damage. I can't rule out a more subtle cold/cracked/fatigued/creep killed joint but it would be a first for me on the SP side of things. In the case of those last ones you need only reflow the solder as well (stick some flux on there , heat it so it melts, maybe add some more if necessary, let it cool down, make sure you have not bridged anything that was not bridged before).
Charge ports of the mini and micro USB variety fail all the time (so much so that I keep a bunch in stock) but the SP is pretty solid. Speaking of which if you are going to replace the port I would suggest you consider a microUSB as it is easier for you in the long run.
Option 2. Bypass the lot. We have seen things made for the original GBA
https://gbatemp.net/threads/integrated-rechargable-gba-mod.465889/ and I recall some nice third party 360 battery packs that connected directly to the pack instead of messing with being official for the 360 controller. You could do something similar. If you are really good you might even be able to retrofit it with one of those wireless chargers they make for phones these days.