Why? If the Wii was configured like the Wii U hardware-wise, this hypothetical Wii would have most likely have sold for like $50 more
It absolutely would not have. Not in 2006.
Nor would it have made any difference, since the non-hardcore audience Wii was targeting didn't adopt HD sets en masse until at least 2010-11 anyway. (Thus, HD would have offered added cost for zero added value.)
And to crank out games that took advantage of such a system would have made the pipeline much slower, as we've seen with Wii U. You think 2013 Nintendo struggling with HD was bad? Wait until you see 2006 "we haven't even fully utilized GameCube's power" Nintendo try their hand at it.
Wii executed on a very specific business model: crappy products for crappy customers (i.e. low-cost, high-margin products for customers who are individually less profitable than traditional customers).
It couldn't have done that by targeting Wii U specs in 2006. That would have required them to put out a high-cost, low-margin product that justified its sticker price by appealing to traditional customer demands like better graphics or traditional controls, which shoves out room for innovation risk (and thus any of the things they did to grab the non-traditional market).
This is basically exactly what we saw happen with Wii U. Nintendo provided a more standardized controller to attract standardized third-party games, and the result was that instead of highly differentiated games (Wii Sports, Wii Fit, Mario Kart Wii) that attract a highly differentiated audience (80 million people who didn't buy GameCubes), Wii U only wound up producing mildly differentiated takes on standardized games (New Super Mario Bros. U, Pikmin 3, Wind Waker HD) that attracted a highly standardized audience (hardcore Nintendo fans who probably bought Gamecubes).