I would like to be able to record a new video clip directly into a track of the timeline while the rest of the project is playing back.

This functionality is built-in to any multi-track audio editor such as Audacity or Ardour.  One important consideration is round-trip latency compensation to account for the time that audio/video from playback takes to be heard/seen by computer output plus the time for audio/video input takes to be grabbed by ffmpeg.  Then when the clip is recorded, it is shifted back a number of frames corresponding to this round-trip latency.  That way the recorded clip will be perfectly in sync to what output the user was responding to.

I would be very happy to implement this, as this feature is something I strongly desire as a musician who likes to overdub multiple instruments into a video. Without this feature, I have to record audio into ardour while separately recording video to a file, and then have to import every track into kdenlive to sync with the video files.  I'm a contributor to MuseScore, so am familiar with Qt and open-source development.