The EZ3 has those too -- it only has SRAM onboard so everything gets to be patched to use it (give or take the handful of games that use the same save type as the EZ3 or have no save/password save). As far as I know every flash cart did it that way (assuming it could save at all) save perhaps the supercards and those are far from clean rom running devices as well.
No, that is indeed how the Supercard worked as well. Everything was patched to save via SRAM instead of maybe EEPROM or FLASH or whatnot, and when playing the game everything was stored in the Supercard's SRAM. The problem was, that SRAM was not battery-backed, so it would be lost a few seconds after shutdown. To actually save permanently, you would need to first save in-game like normal and then do one of three options:
1. Use the button combination to update the game's *.sav file on the SD card with the newly updated SRAM data before continuing on with your game. This is the most convenient option but relies on the Supercard Patcher being able to make the button combo work properly.
2. Power off and then QUICKLY power back on, navigate to the *.sav file, and overwrite it with the newly updated SRAM data. This was known as the "Quick Power Cycle" process.
3. Use the button combination to soft-reset the GBA, navigate to the *.sav file, and overwrite it with the newly updated SRAM data. Essentially the same thing as option 2 but without physically turning off the system.
(4: Alternatively if the game was supported by the patcher, you could also do real-time-saves with the GBA rom using a button combination, just like a savestate on an emulator).
The Supercard miniSD contained a battery which was supposed to be for actually storing the SRAM after poweroff. I don't think this was ever reliable, and even if it did work, you'd have to make sure not to launch a different game by mistake. You'd be best manually doing it with the 3 options above just like normal so it was useless.