Perhaps you could have another camera arranged to focus on a plane immediately within reach, then possibly you could take feeds from both cameras via a mixer to a recorder. Thus a switch to pick which feed is recorded could speed production.
It wouldn't speed production up a huge amount, and ultimately quality is going to suffer with these sorts of systems.
Take for example Chris Gammell's video for Contextual Electronics. He has multiple cameras setup, switches live, and pauses the recording when required. The result is direct MP4 file that is automatically uploaded. No editing required. 15 minutes of content takes 15 minutes to shoot (unless he pauses).
That's great for the sort of stuff he does, but poor for what people expect of my (more polished) videos. Framing suffers, exposure suffers etc, and unless you use HDMI capture from real cameras (not webcams) quality suffers.
My editing doesn't take that long, a 1 hour mailbag takes not much longer than an hour to edit.
Also, unless you are 100% fixed setup to produce the same style of content day in day out, you'd always be dismantling rigs, moving camera, tripods etc around. And it takes longer to set back up to start a shoot. My cameras are always constantly moving around the lab shooting all different sorts of footage. Even having a power cable plugged in to power the camera is a PITA, let alone a big long awkward HDMI cable going off to a PC. It's my idea of a nightmare.
Although I do plan to have such a system like this for putting together quick videos. e.g. a fixed PC with multiple webcams and/or production cams, the Targarno microscope, PC screen capture, monitor screen, X-Split software for live switching, pausing, and recording, and a dedicated wireless mic system to handle the audio no matter what the situation. So I can do live shows as well as record a semi-polished version to upload for people to watch later.