Does the XBox not have an option for doing those updates in the background or when you aren't playing?
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It does but it's not reliable. The update process is also the slowest imaginable other than Windows update. It takes several minutes at times just to discover which games need updates.
It probably does, but it's turned off the majority of time. It's still plugged in, so I wouldn't mind if it updated itself silently, but no, it does not do it when it's turned off.