Can remix also remix the video?

Using remix on a song allows you to make the song shorter with no obvious cuts. Its intended use is to shorten background music to fit the length of a video clip. How about using it in a slightly different way:

I want to use remix on a video of a person singing a song, so that I can make the song shorter, and the video will be cut and joined at the same places as the audio. There will be obvious cuts in the video but it will maintain lip sync with the audio. This is what I do manually when asked to make a shortened/summary video of a concert.

But I can't make remix work on the video. It just ignores the video as if it's not there. Is there a way to make it work?