Do you believe that the child's school progress depends on the teaching performances?

A child's school progress does not depend entirely on the teaching process. The home environment and socio economic background play a vital role as well. However, an educator should be able to evaluate a child's behaviour and take the necessary steps to ensure that the learning process is maximized and smooth.

In many ways yes.  This is because eventhough the child is curious to learn by nature, s/he will also get bored fast and need motivation and incentive.  Part of a teacher's job, I believe, is to spur curiousity into action.  This would be exciting the child enough for him/her to participate and do the work that you are asking to be done.  Children are still very much in touch with imagination so anything that might cause a spark in this regard will be usefull to get children to want to learn more, and hence perform better.