On the subject of the technical expertise demonstrated by Grease: Live, we have a nice example. Carrie Havel, who was the Assistant Director, posted a video to Facebook of a few moments showing what she did during the "Greased Lightning" number, calling out camera cuts. I can't embed a Facebook video here but the folks at Slate can so you can go view it on their site. Like certain magic tricks, some things in television are more impressive when you know how they're done.
And speaking of magic: If you go full-screen and look real careful at the various monitors on display in front of Ms. Havel, you may be able to figure out how they did that amazing transformation of the car during the number.