"How great is the responsibility of a NFP teacher to catechize?"
My initial feeling is that the primary goal of anything we do is to lead people to God. Unfortunately, this is not easily accomplished in a direct manner. I wish it were, then I could just say, "Turn away from sin and be faithful to the Gospel" and the converts would roll in. Ash Wednesday experience clearly teaches otherwise.
In practical experience, a roundabout approach seems more effective. For example, when a woman comes to a crisis pregnancy center, it seems unkind to baldly assert the truth directly, "If you have an abortion you will kill your unborn baby." It seems much more charitable to help her in any moral way possible, in order to encourage her to keep the child. Obviously, one can never deny the truth and must honestly answer that abortion kills a baby if directly asked.
Now, it is the duty of the bishop and his fellow priests to catechize his flock. This should be a constant and expected duty, concerning all areas of life, and occurring primarily during the homily at Mass. I would think that the second catechesis comes from the parents, especially the father. It is the parents' duty to raise their children in the Faith since they best know their children and have the time and relationship to delve into deeper questions. The third catechist would be the individual, especially in the first world where tremendous resources are available with a few mouse clicks. We each have a responsibility to form our own conscience. Experts like NFP teachers or professional apologists seem to come in last in my opinion since they serve primarily as consultants for someone who has already made a moral decision. These experts can offer direction and guidance on particulars, thereby more deeply catechizing someone who has already chosen the high road.
Especially considering the growing trend of using NFP for health reasons, instead of religious ones, our charitable example is far more important than being able to expound on Humanae Vitae (as glorious as that document is). Our charitable example is a better witness than any number of words, but then again I have a Franciscan bent. I see all this as falling under Christ's injunction that without love, we are a gong clashing uselessly.