Some info for anyone interested. I have contacted the Wii Wi-Fi support team, gone through re-enterring all those connection settings on the Wii, updated my router to the latest firmware (not much of an upgrade, but still...), and still have the same failure code: 50100. Because that error comes up quickly, the first layer of support thinks there's something wrong with the connection settings, but I've tried variations with each settings and still get the same error code. So now I have a "callback" request with the Nintendo folks, so some network expert should call within the next 5 days.
Does no one else out there have this problem?