I have written software to do this kind of recording on a laptop, running 4 of the stream itself (different episodes of the same show).
It opened DRM enabled browsers side by side, ffmpeg captured the video from the respective parts of the screen, and each browser's audio was piped into a different dummy output, which ffmpeg also captured of course.
The tech stack was linux, bash, PHP, php-webdriver, Selenium, Firefox, ffmpeg. So yes, this idea absolutely works! That is, until they crank up the DRM so that software screen capture doesn't work.