People always say it's to keep demand for their stuff high, but I dunno. It seems like they could always milk these things for at least twice as long before ceasing production and still do that.
It really bugs me when people say this (and I don't mean you), just because it doesn't make any sense at all. It's such a super simplistic way of looking at supply and demand. Lack of supply doesn't create demand, it creates news stories. I mean, you could argue that those news stories in turn create demand - but who gives a shit unless you can capitalize on that by fulfilling that demand. Nintendo doesn't want people just to see their products as desirable - they want people to see their products as desirable and then buy them right away. Having a shit ton of people wanting a Wii, Switch, or NES Mini doesn't do them any good at all unless they actually turn that demand quantity into sales.
I think this urban legend initially came out with the N64 when there were less consoles available at launch and there were legit chip shortages on some games limiting runs. I think this was a combination of the videogame market growing faster than Nintendo had accounted for and also production problems - but I'm purely guessing here.
Of course there are shortages at every high demand console launch. The logistics of manufacturing consumer grade electronics are probably pretty complicated. Your suppliers roll whole plants or lines over at a time and you have to have a rolling batch order in so you probably book it before all the nitty gritty hardware or design details are sorted, and then it's a race. It's a race to get those details sorted out, a race to get the lines flipped over, and a race to get a test run completed. And then it's a race to produce and ship. Like, I don't REALLY know what I'm talking about here but I imagine it's a pretty high pressure clusterfuck of a situation - We have a release date, we have a date we have our partners plants booked to switch over to produce our product and there's probably not a lot of time in between.
It translates less with the NES mini but the releases are pretty time sensitive. Here's how I imagine it goes down with something like the PS4: You have the R&D mostly specced out and then you can send out to your partners for bidding, or at least pricing. You pick one or approve and then you try to figure out how soon they can roll production over for your product. Lets say they're still producing PS3s or Blueray players, or TVs or whatever. That contract is going for six months more so that's the soonest they can start pumping out PS4s.
Great, lock that date in. Start working on marketing, branding, finer details of design. Crunch some numbers - You have 50% of your manufacturing partners able to roll out consoles in six months, but the rest need 9 to switch lines over. You know you need to launch with actual product on shelves and probably have an idea of how many units you need to have so you can ship SOMETHING to Amazon, Wallmart, EB, and whoever else - because they're going to be taking pre-orders and if you leave them too short you're going to have serious fucking problems with them - and you really do need them and their marketing dollars. So, 50% capacity six months from now, 100% capacity nine months from now. We need a supply inventory of product to sell BUT we have logistics problems with letting that supply build up.
You're in a race with your competitors and technological advancements in general. If Sony thinks the PS4 is going to have a huge impact and wants as much stock as possible to be on store shelves as soon as possible, it would make sense to go into production for, like, a year instead of three months before launching. But maybe that means Microsoft already has their new console out a year earlier and a lot of early adopters and big time console gamers are now tied into their eco system. OR maybe MS launches at the same time BUT they only went into production for three months. Maybe their console now has taken advantage of that nine months extra of development time, enabling them to finalize a GPU or CPU after a die shrink or higher bandwidth memory. Maybe that nine months was just enough to allow them to take advantage of the next generation of CPU/GPU and people are willing to wait to get their hands on the product due to production shortages just because it's just THAT much more powerful.
Beyond that, holding inventory costs money (and this is a huge thing). You don't know if your next console is going to be a Wii or a WiiU. If you overproduce you're not just kicking yourself because you've spent money on this hardware no one is buying, you're kicking yourself because you have to pay someone to store warehouses and warehouses of this hardware no one is buying. You're kicking yourself because you're booked into long term manufacturing contracts that you now need to buy out of because each additional unit being made is costing you money in parts, in manufacturing, and in storage. You're kicking yourself because it's possible retailers are having the same problem and they're now selling your console at a discount which is a detriment to the perceived value of the hardware.
Companies tend to be conservative with estimates for hardware launches, and for good reason. Nintendo is a conservative company in general. I don't think the short supply of some of their products is a marketing gimmick, I think it's a reflection of the fact that this is an experienced company in an extremely fickle marketplace.
And to be honest, as I type this I think the reason behind the NES Classic's termination and the sudden short stock of the 3DS systems is one and the same: The Switch is more successful and more in demand than Nintendo had estimated and they are asking their partners to switch their production lines over from their periphery products in hopes of meeting demand for the Switch. I will now include that in a TL;DR below:
TL;DRA bunch of stuff that's probably wrong about hardware production and a theory I just came up with which outlines how Nintendo may have ended the production run of the NES Classic early and reduced the production of the 3DS family in order shift the focus over to the Switch, which has been received better than they initially estimated.