For simple situations like this, an argument via Venn diagram is probably a good approach. It would be convincing to a reader, though the requirements of a particular course in mathematics may have stricter conditions than "convincing."

For more complicated situations you would need to get comfortable with the written manner of proving things.

To the title question, you may want to know that there is a sometimes-observed distinction between the terms "Venn diagram" and "Euler diagram", where the former shows all overlaps, whilst the latter shows only those which are actually inhabited. There is in general no need to exhibit multiple diagrams, as you have seen, because proving something in maximum generality is sufficient.

Suppose that you have exhibited a diagram that shows what happens in all regions, and now you're going to create a new diagram for the situation where one region is empty. Well, the remaining non-empty regions were in your original diagram, so you already covered them. That explains why the situation is not different in any of the diagrams you created.

The reason you might want to use a diagram that doesn't show all possible intersections is that, for more than three sets, it becomes quite arduous to create such a diagram and the regions can get quite tiny, so it's a matter of clarity.