With all due caution about declaring mission accomplished, today is the third day in a row I could access Netflix via the Wii console with minimal effort.
I do find it interesting that the months-long problem that just got worse with time seems to be resolving itself after I made a label for this issue. I'm assuming a coincidence.
If this holds up, I'll never say another bad thing about Netflix again.
UPDATE: It did not hold up. Now they're just flixing with me.
UPDATE: It worked a day later. And then I missed a day. We'll see if it works tonight, perhaps.