unless u disable the update active IOSs option after using modmii to update your Wii it will overwrite\uninstall cioscorp\darkcorp, and to play backup discs u need to use neogamma or some other loader and can no longer use the disc channel to play backup discs (btw, newer wiis can't play backup discs period).
To play backup discs via the disc channel on your Wii, u can install neocorp (less invasive than darkcorp) or reinstall darkcorp
btw, usb-loading is far superior to disc loading. And u can even rip your backup discs to a usb HDD using either your wii (since your wii is old enough to load backups) or your PC
edit:
re. gamecube games, ModMii would've installed an original MIOS (not cMIOS) unless you enabled the install cMIOS option. This means that you currently can only load original GC games from the disc channel. To play backup games, u need to install a cMIOS (so either enable the option and repeat the wizard, or just download\install the recommended cMIOS from download page 4). Or you can install the cMIOS for dios mios lite and load your GC backups from an SD card, u can use neocorp for this too.