I'm glad you raised this question. I have been thinking about it also. It seems to me that the important idea is how the mean and SD are related to the sketch of a normal curve. I have my students make a sketch and mark the axis with mu and mu +/- a couple SD. Although I don't have learning theory to back me up I think that this is important to do by hand. I also think that the z-score concept is important to do by hand because it is the test statistic used later in the course when you discuss significance tests. Doing this by hand in both places in the course I think helps the student make these connections. I don't think it is important to look up areas in the table or manipulate the table area to get the desired area. Based on these opinions I would want the students be able to make a normal curve sketch and show how to calculate the z-score but then use the calculator to get the area. So I would be using the calculator like an electronic table.
This is of course just one person's opinion. Hopefully your question will get more responses.