After posting this, I found the Nimbus plugin, so I can stream the source video to my Chrome browser on my Chromebook, and have Nimbus record that tab, saving video as a .webm in the Downloads directory on my Chromebook. I then uploaded that file to YouTube. That's a workaround I can live with fine.
It would just be cleaner to just have something like YouTube Live ingest that stream directly and record it into my YouTube Video Manager. I'll check with them.