Most students starting in cryptography either have a computer science background or a mathematics background. In either case, the students are at a fairly basic level.
Computer science students typically do not know abstract algebra, but know basic number theory. This is sufficient for teaching RSA.
Mathematics students often know basic abstract algebra. This is sufficient for teaching RSA. I would expect modules to appear in a second algebra course, not in a first course.
In other words: explaining RSA in terms of a module structure where the ring is the endomorphism ring requires a level of mathematical sophistication that students typically do not have at the point where they first encounter RSA.
However, if this level of mathematical sophistication buys us some understanding from a cryptographic point of view, it could be worthwhile to introduce it, even if it would be additional work for students.
I find that introducing abstract algebraic structures is a great help in many cases, such as understanding index calculus or more practical things like the number theoretic transform.
However, my claim is that in this case abstraction does not buy us any additional understanding, from a cryptographic point of view. (It may very well buy you a better understanding of your second abstract algebra course. In fact, I think this is close to a fairly standard example when teaching modules: an abelian group with scalar multiplication by endomorphisms.)
- It does not help me explain the relation between the two inverse endomorphisms.
- It does not help me explain the relation between factoring and the RSA problem, security-wise.
- It does not help me explain the relation between the algebraic structure and common computational speed-up tricks.
- It is certainly no help understanding the construction of secure schemes based on textbook RSA.
On the contrary, I find that it is much more natural to start with the automorphisms of a cyclic group (which we know is correspond to exponentiation and the Pohlig-Hellman cipher, which may already have appeared as an example of a weak block cipher), explain what we want, observe that exponentiation does not work in a finite field, and then observe that the integers modulo a product of two distinct primes have properties that seem to make it work.
You can also immediately relate the concrete properties of the exponentiation as a number theoretic thing to various factorisation algorithms, which you will also want to study.
Which is a long-winded way of saying that the other two answers are correct.