I understand how to trim a video from either end but how does one cut bits out of the middle? If LR can't do it, is there a plugin that can?
Lightroom's video editing is very basic, Bob, and you can't really cut bits out of the middle. You can use virtual copies to trim the ends and then view each VC in sequence, but that's a horrid workaround.
To do this you really need another program, not a plugin. Your profile doesn't say whether you have CC and therefore Photoshop, but Photoshop's video editing features are surprising powerful and this keeps you in a photographic mentality/skillset. I often feel it's better to deepen one's knowledge of a program one already has. But if you want to get into serious video editing, the Adobe program would be Premiere Pro, or you may be happier with Premiere Elements which is its consumer version.