I guess the prospect of a common underlying OS offers nothing.
No benefits from having a common set of kernel, libraries, apis, etc.
Windows CE has historically been good for embedded devices... only developing (for) it (app or OS level) generally required two different, rather different code bases.
Yes... there will always be differences between platforms (embedded vs desktop, server vs client, mobile vs web, etc)... the more commonalities you can have, the far easier the end to end development story is.
I say "desktop OS" implying it'd be the same as a regular Windows installation (i.e. requiring mouse/keyboard for effective use, and a high-resolution display where the user sits about a foot from the screen). Of course that wouldn't work on a console.
Now, using desktop Windows as the basis of a new console OS (done properly) might work. That's what the vapourware Phantom console tried to do, and to an extent the original Xbox (since it exposed the DirectX APIs).
The thing is, the heart of the Windows OS: the NT kernel, isn't overly suited to gaming console tasks. You don't need a multi-user, multi-tasking kernel in a console. Almost eEverything that sits on top of ntoskrnl is the graphical UI that, too, doesn't make sense for a games console, like the DWM, or the WIMP UI.
Fun little fact: games on the Xbox run in kernel mode.