Gamers. The Wii U right now is really a gamer's console, but that's mainly because they did not manage to sell it to casuals (here meaning people that wouldn't normally play "traditional" video games but might be enticed by the likes of Wii Sports, Wii Fit and maybe a few more accessible games like Mario Kart) and so those "casual friendly" games that were a huge hit on the Wii are not selling well on the Wii U, so they have no choice but to cater to the people that actually bought the Wii U which are mostly avid gamers that are fans of Nintendo.

But right now the Wii U is burdened with its expensive gamepad that people aren't interested in (although I really love the gamepad now and it is actually useful, it was not really a reason I was interested in the Wii U and was more of an obligation), weaker specs than the competition and the lack of third party games. You can tout the Wii U exclusives all you want, but the fact is it isn't just competing with the PS4 and Xbox's exclusive lineups, it's competing with the lineup shared between both of them. You're not just picking Nintendo's exclusives over Halo MCC or Driveclub, than you're picking them over COD, Assassins Creed, GTA, Dragon Age Inquisition and Shadow of Mordor, with PS4 vs Xbox exclusives only coming into it as a tiebreaker once Nintendo has been eliminated.

They could take another gamble on hardware and try to capture the bigger market, competing with mobile etc., but it is a huge risk because I think a failure would essentially kill them in the console business. Inatead they should cater to the gamers. Standard controls, solid specs (depending on when it's released it needs to be either a bit more powerful than PS4 or much more powerful (and in line with PS5 / Xbox X)), strong exclusives and something accessible to third party developers. This gen has shown that there is a no real advantage to built in motion controls, touchscreens or even NFC compatibility - offer all those things too, but as peripherals or compatibility with a handheld. Nintendo wants to be known for focusing on quality gameplay and innovation - well let's see them do that by managing to set themselves apart even on a "standard" console by making great games on it, rather than forcing fans to accept a different kind of console they may not want by making great games on it.

Problem is, that's probably not a very large audience because once you deduct those who prefer PCs or find mobile to be sufficient, you're pretty much left with Nintendo fans. So they already did that. And unless they do compete with MS and Sony (be like the other two, only better at it) or can pull another Wii out of their heinies (unlikely), those people are all they'll have.

How am I the only person who thinks the 50 million or so 3DS owners isn't the most logical choice?

Kids and Nintendo fans..? Nope. The Wii 3 will be the same as the Wii and Wii U before it - the console will be aimed at everybody at launch, regardless of the demographic they belong to.

About half of the 3DS audience is from Japan, and the Japanese aren't into consoles anymore. 

Also 50 million 3DS' sold doesn't actually mean 50 million discreet users either. I own three 3DS units, my brother has two, a lot of families with kids buy more than one per household for each child but they often share the game cartridges. With a console though those parents would likely only buy one console and the kids all would have to share it. 

But you're right to the extent that I think Nintendo's next "console" will basically be just a way to play their handheld games on the television. So to that effect they will be targetted the same audience as it will effectively be the same library. 

How can Nintendo aim their next home console at the handheld market? 

It's not, of course. But both of those groups have become conditioned to a high entry price but then dirt cheap or free software thereafter, they aren't likely to budge in signifcant numbers and are not a healthy group to try and build a console around, anyway.

Since recreating the Wii is a long shot, it seems the most attainable goal would be to aim at becoming the default core console by launching early at a reasonable price with real third party parity this time. Outside of something cute like trying to fudge the very idea of what a home console is, that is.

Aim at the core audience, make their hardware easier to dev for, keep the free online play since that's always a great incentive over having to pay for online over on PS4,XB1 and 360, try courting more 3rd parties, hell even try merging with the 50 mil 3DS base since that's currently bigger than all the other consoles (besides PC).

I wouldn't suggest trying to be exactly the same as PS4/XB1 though, it makes me wonder why should I buy theirs if everyone else is doing the same thing with only tiny minimal changes, I'd rather go for what Ninty does best and having quality games that work instead of being broken on day 1 or having huge downtime issues is something that puts me off entirely when it comes to PS4/XB1.

Also probably try making it a bit more cheaper than the current gen prices along with not trying to force a tablet pad, try going for the regular Ninty gamepad approach and make more games with interesting mechanics and innovations instead of centering it around just the gamepad itself.

The hardcore gamers.