Basically, you have 2 NANDs, sysNAND and emuNAND.
sysNAND is the NAND which stores your native firmware, the one that can be exploited.
emuNAND is a copy of the same NAND that is stored on the sysNAND chip, but it is located on the SD card.
Whenever you boot into Gateway mode, it boots into the NAND which is stored on the SD card (emuNAND). This allows you to not only load the red card and run unsigned code, but also allows you to be on the latest firmware.
My guess is that whenever you boot Gateway mode, it turns the game card slot into a sort of "ODE", which allows it to run the red card, and that's why normal 3DS cartridges do not work in emuNAND/Gateway mode.
And because of the above they needed Classic mode, in order to spoof game cartridges (To run a game without updating) and also region free, and classic mode uses the sysNAND, not the emuNAND. Why? Not sure, but I bet there wasn't good compatability.