Not to mention that when your matrix is 5x5 or more, there aren't even general solutions for roots if for some reason you're still insisting on the matrix->polynomial->eigenvalues route.

What is interesting is obviously personal, and I know you don't mean to denigrate some topics in general, but I want to caution people that characteristic equations and root finding are complex and far from useless. Numerical computation of eigen decomposition starts from finding the roots of the characteristic equations (in altered form), the eigen values, without which there is no computation of SVD. How the behaviors of roots change as coefficients vary are fundamental in control engineering. Newton's method holds up half of numerical optimization. Undergraduates don't have to learn them because others have worked out the details and implemented them in software.

By the way, Cramer's rule is useless for numerical computation, but it is immensely useful in theoretical work. It belongs to the vast body of work dealing with determinants before the rise of linear algebra. Determinant is the only obvious connection to algebra left in an undergraduate's linear algebra course, so I can understand people are turned off by it.