Just tested it, more out of curiosity than necessity:
Within NewPipe (v0.26.1) I am able to record the sound if I activate Screen Record>Audio>Device Audio when starting the recorder.
However, when I try screen recording the YouTube page (and some news pages with video content as well) in Vanadium (120.0.6099.230.0), no sound is recorded.
All done in a secondary, not the Owner Profile.
As I do neiter use the YouTube app nor other third-party screen recording apps, I am unfortunately unable to test them or make a recommendation.