If your Wii reboots after starting a backup disc you should try other dvd media. Had it with a few backups, burned it on a verbatim or tdk - disc, worked like a charm.
I don't think your wiikey is broken, since is stills recognizes the config disc. Also try burning your backups to another dvd media, that should fix it.
(Games burned on crappy media can result in them working fine for a while, but after that without any reason just stop working)
Ps: did you update to 1.9s? If yes you should use the 1.4 config disc.