This video explains how to easily "hide/exclude" Audio of a Stream or Recording using the Software Voicemeeter. We get a Virtual Audio Device that allows us to separate the Audio we want on our Stream from the Audio we do not want. And a Tool that allows us to manage the incoming Audio and send it to our Speakers or Headset.
Any Software that offers the option to select the Audio Output device can be excluded of your Video. (Browser windows for example cannot be excluded)