It's interesting to wonder if Nintendo could've pulled off a 10 year console, but it likely wouldn't have worked nearly as well. The simplicity of Wii Sports working out of the box was one of the bigger draws to the casual crowd, in addition to the strong branding. It might've worked out if there was a console bundle with the remote and Wii Sports, but it probably still would've taken rebranding the GameCube or focusing on the branding of the remote to pull it off.
Not to mention the added benefits of internal memory, built-in Wi-Fi, bigger discs, and upclocking. As much as people like to say the Wii was just an overclocked GameCube, there were several key components added that made the system serviceable even in 2006. Seriously, if you think the Wii was outdated for its time because it was basically a GCN, imagine if it actually was and you couldn't have system-level avatars or connect to the internet without a separate modem. The decision to make the Wii its own thing was crucial, even if they didn't bump the specs that much.