So I guess I'm completely off then, since I thought the battery was needed for rtc games? What happens when the battery runs dry?
General idea.
Reprogramming a game to speak to SD or something is hard, programming a FPGA or something to emulate SRAM, EEPROM and Flash, and all the sub types of each of those, and transparently sort it all behind the scenes is not so bad these days (I believe it is what the everdrive does) but at the time represented something more tricky.
Patching a game to use a different type of save however is very easy, especially on a PC where searching a 32 meg file (practically a lot less than that) for an ASCII string is nothing. On a GBA it is harder which is why we tended only to see it on the DS or more recently with the updates we saw for it.
For whatever reason SRAM was chosen as the thing what gets to be patched to use (hence the term SRAM patch). Nintendo, or whomever developed their carts, actually switched to something called FRAM/FeRAM a year or two into the GBA lifetime and dropped the need for batteries to cover the SRAM. I don't know why flash carts did not follow then or some time later.
SRAM in most forms is noted as needing power to retain the data, hence the battery providing an independent source of it while the cart is not powered.
When the battery dies* the data is no longer held during power down states. If your save is still there it will then not be there for long**. You can try force resetting the cart while it is still on. See the asterisk below but you can possibly just flick the power on and off and have it write back. So yeah you can still save but if you do the normal thing of play a game, save, turn it off and carry on with life it is going to trouble you.
*there is a difference between dies and dies completely. Or if you prefer you have probably had a laptop battery, toothbrush, drill... that was unusable for task but still made a token effort to power lights or move or something. It really is micro amps though to retain data so while it may not even vaguely power a tiny LED it may still be enough for this. Battery draining is not an exact concept, especially at really low draws like this and I don't know what voltage is needed either (if you saw that PDF there would have been nice graphs/plots on there, though the energizer one was more simple than we usually see) but what I did above was good enough for a first pass/back of the envelope thing.
**length of retention upon losing power is often measured in seconds, up to around 20 in some temperatures you might experience (even colder is even longer, hence the cheesy hacker film thing (and sort of real life thing) of spraying RAM with canned air to cool it and lifting it out of the machine to read.