I think this might be a longstanding "bug", but I have also not had any luck on my android using the screen recorder to record device audio from a browser (either chromium or firefox). It used to partially work using the mic to record the speakers, but currently it sounds like it does processing to subtract away the original signal; I hear mostly silence with occasional garbled artifacts resembling the original audio.

Maybe this depends on the site? I have definitely recorded video with audio off YouTube and other popular video sites, on a stock Samsung phone, even yesterday.