How to fix this? Or maybe suggest another audio recording application?
If I understand your problems correctly, I would probably recommend the use of a third-party app here. The free QT X Player requires an extension to capture systems audio along with the screen video and, even then, requires you to manually activate/deactivate the "audio pipeline" each time you use it. Further, it doesn't allow you to mix system and microphone audio sources.
There are a number of apps available that may better serve your requirements. Not knowing the range of your specific needs, I would probably recommend a "middle of the road" app like Screenflick ($29). This app is as simple to use as QT X but adds the capabilities described above that are missing in QT X. Further, it includes limited editing and screen tutorial capabilities found in more expensive apps like ScreenFlow for about a third of the price. (I.e., it is a good starter app that you might find greater use for over time without having to upgrade to more costly software later.) While you can use this app to capture, control, and mix system audio together with your built-in microphone to you screen recordings, I personally prefer to use an external USB mic (Blu Snowball or Yeti) for improved audio commentary. In interested in this app, you can download and test the demo software as found on the developer's web site. There are probably similar and cheaper apps available but my personal experience is limited to Screeny (not suitable for your purpose), Screenflick ($29), SnapzProX ($69), and ScreenFlow ($99).