I meant that instead of using PowerPC, use ARM, which is apparently what the 3DS uses.

Because to make a console backwards compatible, you have to either base your new console around the same architecture as the old one, which is usually very dated by the time you're looking to release the new one (Wii and GameCube), or you throw in hardware that literally makes it two different consoles one box (PS3 and PS2), which makes it quite pricey. Or I guess you could try to go the emulation route (360 and Xbox OG), but that's the weakest option.

Why is emulation the weakest option?

No, they will implement the Power of the Cell processor to achieve hyper realistic graphics. This is needed to achieve their strategic goal of world domination.

Their are tech experts and wannabes on this site who can explain it far better than myself, but it essentially comes down to it being that with emulation, you're taking games that were designed with one console's hardware/architecture and mind and porting it to another's that it has little in common with. It's no different than porting The Last of Us from PS3 to PS4 or any other HD remake, or Nintedo's VC, from a logical standpoint. But the difference is that HD remakes and TLOU remake are done with the point of selling new copies and making more money, so more care is put into them. Same with Nintendo's VC: even though each game is essentially an emulation that required man hours to port over make work, money is being made from them.

WIth emulation done with the purpose of BC, every game from the previous generation has to be done one-by-one for real cash benefit, so when does such a feature start to become too much of a hassle for the console manufacturer? That's why many of the original Xbox games BC compatible on 360 didn't run very well: why put a lot of time and resources into a game you're not making money on for a system you're trying to get people to move on from? That's why MS quickly abandoned BC support for the 360, and why Sony and MS didn't even attempt it with the current consoles.

*for no real cash benefit

I believe ARM is strictly meant for mobile devices, and many of its features would be lost on a home console.

Emulating relatively modern hardware on a completely different architecture is nearly impossible. Gone are the days of simple hardware design like the PS2/PS1/NES/GB/DS. Consoles like the PS3/360 run games using a myriad of complex coding, and emulating that effectively would take years of painstaking optimization, and even then you would run into a ton of problems due to the differences in hardware.

Not strictly mobile no, it was initially designed as a proccessor for business PCs back in the 80s. Of course being a small company IBM, Moterolla and Intel quickly cornered that market. But it was then adapted to be used in embedded systems of all kinds due to it's low cost and power effeciancy. And those qualities plus the flexable licensing terms that let different vendors use the ARM IP and architecture to create custom chips made it pefect for mobile devices of course. But ARM is used in a wide range of devices from Digital cameras all the way up to Servers. The features that make it good for mobile devices being low cost, power effeciant, small die sizes, low heat etc also matches Nintendo's hardware design philosophy perfectly.

It's also worth noting the Jaguar cores that the PS4 and XBOne use for their CPUs is actually designed for mobile devices like tablets and netbooks, as well as microservers and embedded devices. It's a very low end x86 part designed to compete in part with the high end ARM parts.

So would it be possible for Nintendo to make an ARM console as powerful as the PS4 for $300 with backwards compatibility to the Wii U and/3DS eshop by winter 2016?

Backwards Compatibility is just a term that describes emulation of predecessors.

Emulation can range from running entire virtual operating systems to excuting simple predecessor instructions.

There are two types of emulation, software and hardware. Hardware emulation requires the actual components to be in the system, this means you can A) build the newer model using similar technology, which holds the system back if the architecture is too outdated or B) include the old architecture along with the new one, more expensive. Software emulation requires architecture that is alot stronger then hardware its emulating because it virtualizes the hardware in its own, and that virtual hardware runs the software. Typically, emulators run the whole OS, but well optimized emulators can run barebones and stripped OS. E.G, my phone can emulate some PSP games at native resolution and framerate despite only being maybe 5 times stronger.

Long story short, hardware emulation is the only viable online solution for emulation and subsequently BC, streaming can do the hardware emulation by running it on a more powerful server, but their is the issue of network dependency.