There are other benefits as well. For example, models of set theory within the given one ("Inner models") can be discussed with almost no prerequisites from mathematical logic. A more mundane application: The algebraic closure of a field really can be constructed by adjoining roots of polynomials one at a time, transfinitely, with no overarching set required. This looks so suspicious that no major textbook that I know of uses it, but it can be done with a little (standardized) care. This example (and others, some standard and some not) will be discussed in the course.

By the way, the three innovations here (early use of ordinals, classes, and replacement) really only produce two obstructions. Without classes, the axiom of replacement is a killer--Halmos [2] relegates it to section 19, under its alias, substitution. With classes, it's a two-clause sentence: If the domain of a [class-theoretic] function is a set, then its range is a set. There are subtleties, of course; that's part of the course. References:

1. Devlin, Keith. The Joy of Sets, Springer 1993.
2. Halmos, Paul. Naive Set Theory, Springer 1970.
3. Just, Winfried, and Weese, Martin. Discovering Modern Set Theory, AMS 1996.
4. Vaught, Robert. Set Theory--An Introduction, Birkhaeuser 1985.

Prerequisite: One quarter of any beginning graduate mathematics course. (This is for mathematical maturity purposes, and resembles the reasoning behind the calculus prerequisite for linear algebra.)