Sorry, but you heard incorrectly: AE doesn't do screen grabs of video. You would need a third-party application such as Camtasia to do it. You could then track the resulting video in AE.
I found it! It is possible.
In the Paint Tool Menu you have the Option "Duration".
This can be set to "Write On".
Every Brush Stroke will be captured
Motion Sketch is far differerent than a screen recording. Hence the confusion.
Your are right.