I agree. I don't think it's going to be anything to get excited about.
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I'd travel and shoot with whatever working camera I had.
This. I used to get mostly great photos of plays even with my relatively ancient XTi. In my experience, it isn't the camera that makes or breaks stage photos, but rather whether you shoot in RAW and whether you know how to use the camera in full manual mode so that your shots aren't consistently blown out by the automatic exposure thinking it needs to bring out the details of the black floor rather than the bright highlights that just happen to be people's faces.
I think a 6D would do a stunning job, and I'd be surprised if the better autofocus of a 5Dmk3 or a 1DX would make much difference in such a controlled environment. Just my $0.02.