Even grabing the camera to see if the camera tracks doesn't work as well. Did I do something wrong?
Just take track to modifier and pull it under the child of modifier and then click on Z and Y again (to reset it or something) and it works, btw it is a really nice concept.
Your curve cannot pass through that green line, that causes that rotation you see... you need to adjust it a little if you don't want to change the whole camera setting a little bit like this:
In this case it works right