The GameStop employee was incorrect.
Nintendo shut down some online functionality last year. Gone was their promotional Nintendo Channel that hadn't seen any new content for months prior, messaging and Mii transfers, the Everybody Votes and Check Mii Out channels, and the news and weather channels. Essentially if it involved Wii Connect 24, it was eliminated.
Retained were the more essential online services. Namely, the 3rd party video streaming apps like Netflix, the Wii Shop, online multiplayer in still active games, and the internet channel (Which I don't see how they could ever disable).
Then several weeks ago, Nintendo shut down their online multiplayer in games like Mario Kart Wii due to the goings on with their online multiplayer contractor, GameSpy. Left after that are the video apps like Netflix, the remaining remnants of 3rd party multiplayer enabled releases like Activision's Call of Duty line, the Wii Shop, and the internet channel.
I did the system transfer and downloaded several games back in April just fine. Still intend to grab Super Hang-On though, so I'll have to log back in soon and pick that one up and then call it quits after 7 and a half years.