This is a complete myth. The Wii actually peaked later and sold well for longer than all of Nintendo's other home consoles; except the NES, which had really weird release schedules.
1. Wii's peak month was December 2009, three years after launch. It sold a whopping 7.4 million units globally, the highest of any home console, and third highest overall - as the DS beat it twice with 7.5 and 7.8M sales.
Comparatively speaking, the Wii had a healthier late-life than all of Nintendo's other home consoles... Except the NES, but a lot of that had to do with the very slow introduction it had throughout the world. Many European markets saw no official release until the same time as the SNES.
I had a thread from a while back (can't find it now) with a discussion about how, instead of releasing the Wii U, Nintendo may have been more financially successful had they put their efforts into a Wii HD, or even continued supporting the Wii until they were ready for a console like the Switch -- which, in reality, is probably where they would have gone had all the tech been there by the end of 2010. The Wii U was a stopgap, similar to the Gamecube.
The myth is that Wii sold well for three years and then next to nothing afterwards. I am also looking at calendar years to determine the peak, not individual months. Additionally, I said sales curve which isn't something you address with your third point; Wii may have sold above 2m units longer than the SNES and N64, but the Wii also peaked higher than both of those systems. So this advantage of the Wii doesn't result in a different sales curve.
The important point is that Wii would have sold much more than 101m if it had been treated like a winning system. Your final paragraph goes in that direction and I definitely agree that keeping the Wii around for ten years would have been better than making the Wii U. Where you are wrong is that neither the GameCube or Wii U were stopgaps; they were entirely different concepts from Wii and Switch, and had GC and Wii U been successful, neither the Wii or Switch would have been made. The existence of the Wii U is evidence that Nintendo would rather make GCs than Wiis or Switches, because there was absolutely no business sense in making the Wii U.