For anyone interested, here's what I found when comparing the standard and family-edition Wii drives.
This is the underside of the drive lid from a standard Wii from 2006/2007:
Here's the same thing on a Family Wii:
The only meaningful difference is the extra bit of metal in the green circle. This stops GC discs from getting as far as they need to in the drive. It also ensures that GC discs don't get away from the roller at the front of the drive so they can be ejected.
This is a white plastic piece in the top-right corner of a standard Wii's drive (with the lid off):
This is the same thing in a Family Wii:
When this piece of plastic is moved in response to a disc being inserted a certain amount, the drive latches the disc down onto the spindle motor. The Family Wii has basically removed the section of solid plastic necessary for this latching to occur with GC discs. When a GC disc is inserted, the roller at the front keeps trying to push the disc forward, but the metal bit from before indirectly holds it back and the disc isn't getting latched. After several seconds of the disc not getting latched, the drive ejects, tries to re-load the disc, then ejects and gives up.
Now the fun part!
Here's the extra metal bit from earlier bent back to match the standard Wii:
Same thing, different angle (and there's plenty of clearance, so leaving it like this is fine):
Finally, the Family Wii's plastic piece modified to approximate the standard Wii's design:
I used a razor and glue gun to modify the plastic. It's probably a lot easier to just melt the plastic or use other proper cutting tools that I don't have. Don't try this at home if you value the life of your Wii.
After this, the Family edition Wii loads GC discs without an adapter. Hooray! Maybe a similar modification will be possible for the Wii U if its drive turns out to be able to read GC discs with an adapter.