It appears that I have an unmoddable Wii.
I bought it at Gamestop in PA on April 26th, the serial# starts with LU566.
I was hoping that I had purchased my Wii from a previous mod friendly batch. But, when I opened it up to put in a D2Ckey with a Wii-Clip I was dismayed to see GC2-D2C floating on the chip with no numbers below it. Research from here told me that not all of these chips are a no go, so I installed the chip anyway. Game disks work just fine, but backups come up as "unreadable disks".
I do need to follow up by saying that I have never successfully modded any Wii's or burned a Wii backup. I have however done a decent amount of research on this site and others to be fairly certain that I have done everything right (on that note: thanks to ModderMan and all of the other helpful people on this site). I guess that I will wait to see if a new chip or any other solutions come out and in the mean time convince a friend to let my try my hand at modding his Wii.