Light weight, free way to record a video with a blurred background using Mac webcam?
The easiest approach to attain your goal is to use a video calling application in combination with screen recording.
For the video calling application I suggest Jitsi as it works in your browser, but you can also look into any other popular application like Teams, Zoom or Skype.

Go to meet.jitsi.com to start a video call in your browser. For other applications follow the steps they need to start a video call.
Configure the calling application to use your camera and microphone.
Configure background blurring.

In Jitsi you can go to the three dot menu in the lower right and select &quot;Blur my background [BETA]&quot;.
Here are instructions for other applications: Teams, Zoom, Skype


Now you should have a screen, showing live what is being recorded by your camera.
Open Quicktime, and follow these steps to record your screen, in particular the window of your video calling application.

This will generate a recording of your screen, and as that screen shows your webcam with a blurred background you will have a video of yourself with a blurred background.
Note that some video calling applications have a built in recording option, allowing you to skip step 5.