I'd actually urge you to reconsider your stance on the Courant-Fischer theorem. I think it's very fitting for your course for at least 3 reasons:

1. It shows that the eigenvalues are "there for a reason" - they are not just random numbers who happen by luck to solve the eigenequation, they solve optimization problems that make sense. Now, you may want to prove just the extremal characterizations - they are much easier - and to state the more compicated general for without proof.
2. They are very useful in obtaining bounds on eigenvalues.
3. They keep popping up in applications. To say nothing of the physical problems that motivated the Rayleigh form in the first place, it comes up in spectral clustering and in the derivation of the Fisher linear discriminant - two applications that most engineering students will be sure to appreciate.