I've noticed this feature, but it doesn't allow you to record audio with it too (mic or system sound). Am I missing something/is there a good way to do this?
I can't find the option for audio + screen recording and your picture doesn't show it.
After you click the "New Screen Recording" button a small rectangular window appears. Next to the Record button there is a drop down menu where you can indicate sound input source, as well as things like "Show Mouse Clicks". I recently came across this Quicktime tool. It's really nice, you can even do some simple movie editing with it (trim clips, combine clips etc). I use it to produce demo videos.
in the drop down where you select the iPhone as video source you can also select the iPhone as audio source. I'm not sure if it records the audio output of the iPhone or the microphone input of the phone.