I found an old Gamecube at a second-hand store going really cheap. The catch was, the disc would spin up, and then stop again (with no disc message/indication). If the drive detects no disc, it won't spin the motor. So if your laser is dying, and it cant see anything, it wont spin the disc. So it is possible that the laser is dying/almost dead, but there ARE guides out there that may help with this; just google "gamecube laser calibration" and hit the first link.
My Wii one day all of a sudden stopped working (potentially like yours). It would turn on, and give a black (black+sync) signal to the TV, but no sound/picture. Also, the drive light eventually flashed, and the Wii-motes would not connect or sync. As your drive light is working, i assume the basics are still alive (CPU/motherboard). This is good. After much messing around, I stumbled across another Wii (for spare parts) and swapped the Bluetooth module into my Wii. This fixed the problem, it boots up fine, and is working perfectly! Apparently the WiFi module can also cause a similar problem, so you could try swapping the bluetooth and/or WiFi modules from another Wii to check. If it IS the bluetooth module, i know www.dealextreme.com sells them for about USD$8.