Nintendo, since the GameCube, had been using the PowerPC (made by the guys at IBM) architecture, which was used by old Mac systems back in those days (Mac OS 9 and below. Apparently, the Xbox 360 used it as well). But the problem is, Ninty's marketing, and the architecture.
Because, see, not only Ninty, as we all see, has the bad habit of not marketing correctly, but worst, for third-party developers, PowerPC is a living hell to deal with. See, most of the x86 based processors used a certain way to process data. The "Endian" thingy you maybe have heard.
Let's say that a x86 processor will represent a 8-bit number like 15 like this: "00001111". Well, most of the PowerPC processors will display it like this: "11110000", because, the smallest bits come first, which means that it was tricky to write or port libraries or games who used bitwise arithmetic (Instead of working with numbers, they worked with bits. Processors prefer bitwise stuff since it's easier to translate to machine code/process than numbers. On small projects, you won't notice a change, but for big games, it can be a game changer).
But not only this. Years have passed, and quite a few flaws have been discovered in the PPC architecture which made it inferior to the x86/IA-32 processors (x64 were too rare yet). But, when the x64 came, well, the PPC was totally deprecated, since most of the people switched to x64/x86 (code was way easier to make for those, not to mention that any well-written x86 code will work on a x64 processor), and unfortunately, most of the libraries with it didn't enjoyed ages without maintenance.
And sadly, Nintendo's mistake is the same: By choosing to go different, they always choose PowerPC architectures. Of course, since their last console was another PPC-based one, they can easily port their stuff without too much hassle. But for third-party devs who write/breathe/live/see x86/x64 everyday with PlayStation systems or computers, that's a pain, sicne they have to rewrite all of their code! (Not to mention the low frequency on PPC processors). Of course, Apple eventually dropped PPC (not sure, but apparently that's the case), in favor of the infamous x86/x64 for most of their computers nowadays, but, there still are a few companies who use PPC processors. Which means that making games for Nintendo's consoles (Not the handhelds! The consoles) is a totally pain for most of the people who are adapted to write code for x86/x64 processors. Hence why Nintendo's consoles' hardware always was subject to many debates, both in the video game community, and in the programming community.
Note: I'll end the architecture debate here.
