I believe you're correct - Xbox runs a modified Windows 20000, and subsequent versions I'm not certain on but I know it basically stands up a modified Hyper-V and the parent partition is the interface OS with games booting whatever kernel they were built for inside a VM
Well, it is true, but Windows did run BE on the original MIPS R3000 platform. And only on the R3K[0]. The CPU architecture flag is still defined on modern Windows as IMAGE_FILE_MACHINE_R3000BE. There's an early test build of Win3.1 + GDI somewhere that runs on this platform.
The actual first release of WinNT 3.1 only supported MIPS R4000 and higher, I think. In little endian mode.
[0] I know the Xbox used a modified NT kernel, I've seen claims that the Xbox 360 also was, which would make it the second NT system to run big endian. Not familiar enough with sources better than wikipedia to trust that it actually was.