I've been on this theory for the past few years that Nintendo's audience (which includes both "fans" and "general consumers") care just as much about hardware as they do software, despite the common misconception that that isn't the case. The difference is: they care about hardware convenience, more so then necessarily "power" (although power can factor into how convenient a product is). OK, that is simplifying things way too much because every popular console has it's own brand of "convenience" to the consumer, after all at the end of the day all consoles are about convenience. But the perceived convenience of something like a Playstation is about being as powerful as possible while still being cheap and mass produced. That form of convenience doesn't land for Nintendo or their consumerbase as well, Nintendo has basically positioned themselves so that there is no way their products can be convenient to their audience without being different. There is absolutely nothing convenient about a console where the controller is a tablet with features that nobody uses, with that tablet and those features jacking up the price, with most of the marketing being geared towards an audience which has all but left the traditional gaming industry by now. There is nothing convenient about a mini cube that, no matter how good the games are on it, is playing the same game as two other competitors with much deeper marketing pockets that have more third party support or innovative features. I guess you could say the Gamecube losing to the Xbox had less to do with convenience to the consumer and more to do with Microsoft's deep marketing pockets, but in general Xbox and Gamecube being so far behind the Playstation 2 showed how little the actual hardware had to bring to the table in terms of convenience to the user. There is nothing convenient about a console that is more powerful but has a significantly smaller selection of games than the competition and also has those games at prices that are often much higher than CD games as well (and I mean MSRP for virtually all games, not just "Nintendo never does sales on their first party games!").
Nintendo's brand of convenience is about form factor, "uniqueness" (not just in product but in position in the market), and cheap prices. That's basically how you make a modern successful Nintendo system. And it's why the Wii U, Gamecube and N64 either failed or didn't live up to their standards. It's also why the 3DS needed a big ass push behind it in order to get it lifted off the ground. DS-type games and 3D effects weren't going to carry it. For all intents and purposes it wasn't actually as convenient or appealing to the consumer as it probably should have been, which is a big reason why it didn't hit the 90m+ mark I think.
Software sells hardware. Hardware also sells software. Go figure.
The key to understanding Nintendo's appeal is to know about what put Nintendo on the map in the first place and that means to examine gaming of the 1980s, so long before PlayStation showed up. I mean, for the most part you are on the right track with your post, but there are a few pieces missing.
What consumers associate Nintendo with is games that aren't wasting their time. Nintendo first made games for the arcades before they created their own console. The business model of the arcades was that game developers had to convince consumers right away that the game is worth playing, because otherwise the consumer would simply move over to the next cabinet to put their coins into. The convenience you talk about in your post could be said that a fun game doesn't need more than two or three buttons to be played; that in turn makes long tutorials redundant which translates to less time wasted.
The first time Nintendo lost in the console business was with the N64 which had a controller with a lot of buttons and the shift to 3D games made games more bloated. In other words, it took more investment to get into a game and the intensity of the games was lower. If you look at the example of Super Mario games, the SMB series had the player constantly be confronted with enemies and bottomless pits, so decisions had to be made by the player on the fly. On the other hand, you have Super Mario 64 where you can simply walk around enemies and deadly hazards are way less present. This is what I mean when I say lower intensity.
I'll insert a paragraph here to address the pressing question why more complex games succeed on Nintendo hardware as well, such as Breath of the Wild on Switch. The NES had action-adventure and RPG hits with IPs like The Legend of Zelda, Final Fantasy and Dragon Quest, so it's nothing out of the ordinary that Nintendo's consumers also seek games that require a much higher time investment than titles that you could put into an arcade and have them succeed. But it's important to recognize that "easy to learn, hard to master" games have historically always been a bigger draw for Nintendo consoles.
Back to Nintendo's consoles, the GC took small corrective steps because Nintendo had recognized that the N64 controller had taken things too far. That's why the GC controller has this arrangement of differently shaped buttons and why the A and B buttons were meant to stick out as the main buttons with their green and red colors. Of course in hindsight it's an easy call to make that that didn't help Nintendo's fortunes.
That's why Nintendo emphasized in their Wii presentation that it's important to create a new starting line. With the Wii they were fully committed to this idea; a Wii Remote held sideways resembled an NES controller while it was also capable of motion and IR tracking. The console's design conveyed that video games were quick to pick up again and that brought back former Nintendo consumers in huge amounts.
As you correctly identified, the Wii U's Gamepad does not send such a message at all. It was total regression on Nintendo's part.
Switch's Joy-Cons are able to largely fulfill the roles of Wii Remotes, because a single one Joy-Con held sideways doesn't present players with a lot of buttons, nevermind the option for motion controls. It makes multiplayer games a lot more inviting than the Wii U did and for the first time in a long time, Switch is a console that comes with two controllers right out of the box.
When you look at Nintendo's handhelds, they maintained the essence of not wasting players' time for a long time by virtue of technological limitations, i.e. a handheld could not feasibly feature 3D graphics until much later than home consoles. While the DS was capable of 3D games, the actual bread and butter on the system were still 2D games. It was only with the 3DS that Nintendo could really go wrong and that's exactly what they did.
One last thing about Switch: A big reason for its success is that Nintendo doesn't dictate how consumers are supposed to use the device. With both the Wii and the Wii U there were plenty of instances where games didn't provide controller options when they feasibly could have. Switch provides a lot of flexibility and convenience, both in controller options and the ability to play games wherever the player chooses.
You are a rare breed, Alchy. You think for yourself. I don't see that often. Just wanted to say that.