Quicktime Player is an application built into the Mac operating system by default. It was designed to capture a Movie recording using the built in microphone and camera. It was also designed to capture a Screen recording from an external device or internet streaming and is used by millions of people around the world. You can download or record anything you want but it is what you do with that material that matters. I won't go down that road but general knowledge and common sense should prevail.
This forum is for helping with technical and user issues for anyone who needs it.
A screen recording will capture any video on you screen and audio from the built in microphone. Although this method works, the audio is limited in quality as it only captures sound from the built in speakers and is subject to other sound artefacts in the nearby surroundings. To gain optimum sound quality, we need to capture the computer system audio and bypass the internal microphone. This is, and Apple have made it a very simple process. By installing a so called third party application, we can achieve very good sound quality.
Google and download an application called 'SoundFlower', install it and restart your computer. To check that it is installed, open System preferences/Sound and it should appear in the open window along with Internal Microphone and Internal Speakers.
It is clear that you know how to do a New Screen Recording, so open the stream or video that you want to record. In the Record Window that opens, next to the record button is a little down arrow, click and hold it and a dropdown menu will appear. Select SoundFlower and proceed. You may need to check in system/prefs/sound and select SoundFlower as Input and/or Output.
Interestingly, Apple produce a paid version of Quicktime from the App Store called RecMaster. If you decide to try this application, it will suggest and direct you a better Audio Capture Device called, 'Aunboxsoft Audio Capture Device'. It is free. This in my opinion is better than the default QuickTime app. and can be downloaded in place of SoundFlower. RecMaster is more versatile, works, exactly the same but with a better range of file formats to suit other applications.