As I have been slowly moving towards Linux for my Media Center usage, this is one aspect I had not yet tested in my requirements.
Thank-you for raising the issue.
To the point, as I see it, as a plain, vanilla user on the up-to-date Linux Mint (using Mate window manager) laptop here, I have found that Media Center works well for my needs.
But those needs have not yet included an analysis of how well Theater mode works for my needs.
So, based upon your comment, I pressed Ctrl-4, and went to the Exit menu pick.
When I went to the System Sleep, Shutdown & More menu pick, I did not see a shutdown option to select.
OK, for me, that prevents me from using Media Center in my entertainment center.
While I have been testing Media Center on Linux as a replacement for Media Center on the Windows 10 PC in the entertainment center (and, honestly, it's been all sunshine and roses up until this thread), I had not yet faced the shutdown issue.
So, for me, and my testing of Linux, it is now a full stop. I seem to be stuck with Windows.
And, I want to add, on this Linux Mint laptop, where Media Center does not show me the previously noted Shutdown prompt. I can issue, as a user, the command
shutdown -P now
and the laptop shuts down.
So, there does not seem to be a need for admin privileges to shutdown the computer, at least from a command line.