I had it backwards! I'm no expert on Nintendo things. That Wii I was thinking of, the Wii Mini (originally sold in Canada) didn't have WiFi at all.
Reading more on this, it renders some games absolutely useless. However, the titles lead me to believe they weren't really popular games. Nintendo has a whole slew of games in an unwanted category less appealing to collectors than sports games! Fortunately, it doesn't make the thing completely useless:
http://www.forbes.com/sites/amitchowdhry/2014/02/27/nintendo-to-shut-down-wi-fi-connection-service-for-wii-and-ds-in-may/