Why doesn't it save games?

It doesn't depend from the GBA itself. Most cartdrige, if not every cartdrige used in the GBA and the Game Boy (Classic/Color/Mini...) has a small battery, the round one used in clocks soldered inside, used to power the internal memory reserved for the saves.

When the battery, after a very long time (it depends: go look on Google for "Pokemon Gold". Actually almost no one of the old Game Boy cartdriges can hold saves anymore without external intervention), dries out, you lose the ability to save.

I don't know if the battery inside is rechargeable: for example I own a Pokemon Gold Cartridge that I had to hack open, "Desolder" (with a sharp blade to avoid placing an heated soldering iron on a charged battery: a recipe for disaster) the batter, and repack the internal board with a new battery and lots of duct tape.

If you truly need to recover that cartdrige, and leaving it in a charged up GBA for long doesn't do the trick, you may consider this idea